Subgroups — #30859359

Order 17152.

866 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863443	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22568771	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863421	C₂	1	1	yes	maximal, characteristic
5	#22568626	C₂	1	1	yes	maximal, characteristic
6	#22560985	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560984	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560983	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560983	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560982	C₂	1	1	yes	maximal, characteristic, complemented
11	#22560981	C₂	1	1	yes	maximal, characteristic, complemented
12	#22560981	C₂	1	1	yes	maximal, characteristic, complemented
13	#22559934	C₂	1	1	yes	maximal, characteristic
14	#341×D₆₇	C₂	1	1	yes	maximal, characteristic
15	C₆₇×#1950	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#11761611	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×D₆₇	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×D₆₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×D₆₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×D₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
22	C₂×#14524000	C₂²	1	1	yes	characteristic
23	#16863443	—	2	4	no	—
24	C₂×#14523979	C₂²	1	1	yes	characteristic
25	C₂×#14523979	C₂²	1	1	yes	characteristic
26	C₂×#14523978	C₂²	1	1	yes	characteristic
27	C₂×#14523978	C₂²	1	1	yes	characteristic
28	C₂×#14523978	C₂²	1	1	yes	characteristic
29	C₂×#14523978	C₂²	1	1	yes	characteristic
30	#108×D₆₇	—	2	4	no	—
31	#16863421	—	2	1	no	—
32	#16863421	—	2	1	no	—
33	#16863421	—	2	1	no	—
34	#16863421	—	2	1	no	—
35	#16863421	C₄	1	2	yes	—
36	#16863421	C₄	1	2	yes	—
37	#16863421	C₄	1	2	yes	—
38	#16863421	C₄	1	2	yes	—
39	#16863420	C₂²	1	1	yes	characteristic
40	#16863419	C₂²	1	1	yes	characteristic
41	#16863419	C₂²	1	1	yes	characteristic
42	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
43	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
44	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
45	#16863353	—	2	1	no	—
46	#16863353	—	2	1	no	—
47	#16863350	—	2	1	no	—
48	#16863350	—	2	1	no	—
49	C₄×#11761612	C₂²	1	1	yes	characteristic
50	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
51	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
52	#16863161	C₂²	1	1	yes	characteristic
53	#16863161	C₂²	1	1	yes	characteristic
54	#16863160	C₂²	1	1	yes	characteristic
55	#16863160	C₂²	1	1	yes	characteristic
56	#16863160	C₂²	1	1	yes	characteristic
57	#16863160	C₂²	1	1	yes	characteristic
58	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
59	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
60	#423×C₆₇	C₂²	1	1	yes	characteristic
61	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
62	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
63	#341×C₆₇	C₂²	1	1	yes	characteristic
64	#341×C₆₇	C₂²	1	1	yes	characteristic
65	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
66	C₂×#11761612	C₂³	1	1	yes	characteristic
67	C₂×#11761612	C₂³	1	1	yes	characteristic
68	C₂×#11761612	C₂³	1	1	yes	characteristic
69	C₂×#11761612	C₂³	1	1	yes	characteristic
70	C₂×#11761611	—	2	1	no	—
71	C₂×#11761611	—	2	1	no	—
72	C₂×#11761611	—	2	2	no	—
73	C₂×#11761611	—	2	2	no	—
74	C₂×#11761611	—	2	1	no	—
75	C₂×#11761611	—	2	1	no	—
76	C₂×#11761611	C₂×C₄	1	2	yes	—
77	C₂×#11761611	C₂×C₄	1	2	yes	—
78	M₁₆×D₆₇	—	2	2	no	—
79	M₁₆×D₆₇	—	2	2	no	—
80	M₁₆×D₆₇	—	2	2	no	—
81	M₁₆×D₆₇	—	2	2	no	—
82	M₁₆×D₆₇	—	2	2	no	—
83	M₁₆×D₆₇	—	2	2	no	—
84	M₁₆×D₆₇	D₄	1	4	yes	—
85	M₁₆×D₆₇	Q₈	1	4	yes	—
86	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
87	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
88	C₂×C₈×D₆₇	—	2	1	no	—
89	C₂×C₈×D₆₇	—	2	1	no	—
90	C₂×C₈×D₆₇	—	2	2	no	—
91	C₂×C₈×D₆₇	—	2	2	no	—
92	C₂×C₈×D₆₇	—	2	1	no	—
93	C₂×C₈×D₆₇	—	2	1	no	—
94	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
95	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
96	#14524000	C₂×C₄	1	1	yes	characteristic
97	#14524000	C₂×C₄	1	1	yes	characteristic
98	#14524000	C₂×C₄	1	1	yes	characteristic
99	#14524000	C₂×C₄	1	1	yes	characteristic
100	#14524000	D₄	1	1	yes	characteristic
101	#14524000	D₄	1	1	yes	characteristic
102	#14524000	D₄	1	1	yes	characteristic
103	#14524000	D₄	1	1	yes	characteristic
104	C₂×Dic₆₇⋊C₄	Q₈	1	1	yes	characteristic
105	C₂×Dic₆₇⋊C₄	Q₈	1	1	yes	characteristic
106	C₂×(C₂×D₆₇)⋊C₄	Q₈	1	1	yes	characteristic
107	C₂×(C₂×D₆₇)⋊C₄	Q₈	1	1	yes	characteristic
108	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
109	#14523979	—	2	2	no	—
110	#14523979	—	2	2	no	—
111	#14523978	—	2	2	no	—
112	#14523978	—	2	2	no	—
113	#14523978	—	2	2	no	—
114	#14523978	—	2	2	no	—
115	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
116	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
117	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
118	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
119	Dic₆₇⋊C₈	—	2	1	no	—
120	Dic₆₇⋊C₈	—	2	1	no	—
121	Dic₆₇⋊C₈	—	2	1	no	—
122	Dic₆₇⋊C₈	—	2	1	no	—
123	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
124	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
125	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
126	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
127	#14523954	—	2	1	no	—
128	#14523953	—	2	1	no	—
129	#14523953	—	2	1	no	—
130	C₄×C₆₇⋊C₈	—	2	1	no	—
131	C₈×Dic₆₇	—	2	1	no	—
132	C₈×Dic₆₇	—	2	1	no	—
133	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
134	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
135	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
136	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
137	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
138	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
140	#108×C₆₇	—	2	2	no	—
141	#108×C₆₇	—	2	2	no	—
142	C₄⋊C₈×C₆₇	—	2	1	no	—
143	C₄⋊C₈×C₆₇	—	2	1	no	—
144	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
145	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
146	#97×C₆₇	—	2	1	no	—
147	C₄×C₈×C₆₇	—	2	1	no	—
148	C₂³×D₆₇	C₂×Q₈	1	1	yes	characteristic
149	C₂²×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
150	C₂²×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
151	C₂²×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
152	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
154	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
155	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
156	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂×C₄×D₆₇	C₂×Q₈	1	2	yes	—
158	C₂×C₄×D₆₇	#41	1	2	yes	—
159	C₂×C₄×D₆₇	#41	1	2	yes	—
160	#11761612	—	2	1	no	—
161	#11761612	—	2	1	no	—
162	#11761612	—	2	1	no	—
163	#11761612	—	2	1	no	—
164	#11761612	—	2	1	no	—
165	#11761612	—	2	1	no	—
166	#11761612	C₂×D₄	1	2	yes	—
167	#11761612	C₂×Q₈	1	2	yes	—
168	#11761611	—	2	4	no	—
169	#11761611	—	2	4	no	—
170	#11761611	—	2	4	no	—
171	#11761611	—	2	4	no	—
172	#11761611	—	4	2	no	—
173	#11761611	—	4	2	no	—
174	C₂×C₆₇⋊C₈	—	2	1	no	—
175	C₂×C₆₇⋊C₈	—	2	1	no	—
176	C₂×C₆₇⋊C₈	—	2	1	no	—
177	C₂×C₆₇⋊C₈	—	2	1	no	—
178	C₂×C₆₇⋊C₈	—	2	1	no	—
179	C₂×C₆₇⋊C₈	—	2	1	no	—
180	C₂×C₆₇⋊C₈	—	2	1	no	—
181	C₂×C₆₇⋊C₈	—	2	1	no	—
182	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
183	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
184	C₈×D₆₇	—	2	4	no	—
185	C₈×D₆₇	—	2	4	no	—
186	C₈×D₆₇	—	2	4	no	—
187	C₈×D₆₇	—	2	4	no	—
188	C₈×D₆₇	—	4	2	no	—
189	C₈×D₆₇	—	4	2	no	—
190	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
191	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
192	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
193	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
194	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
195	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
196	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
197	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
198	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
199	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
200	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
201	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
202	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
203	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
204	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
205	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
206	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
207	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
208	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
209	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
210	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
211	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
212	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
213	M₁₆×C₆₇	—	2	1	no	—
214	M₁₆×C₆₇	—	2	1	no	—
215	M₁₆×C₆₇	—	2	1	no	—
216	M₁₆×C₆₇	—	2	1	no	—
217	M₁₆×C₆₇	—	2	1	no	—
218	M₁₆×C₆₇	—	2	1	no	—
219	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
220	M₁₆×C₆₇	C₂×Q₈	1	2	yes	—
221	C₂×C₈×C₆₇	—	2	1	no	—
222	C₂×C₈×C₆₇	—	2	1	no	—
223	C₂×C₈×C₆₇	—	2	1	no	—
224	C₂×C₈×C₆₇	—	2	1	no	—
225	C₂×C₈×C₆₇	—	2	1	no	—
226	C₂×C₈×C₆₇	—	2	1	no	—
227	C₂×C₈×C₆₇	—	2	1	no	—
228	C₂×C₈×C₆₇	—	2	1	no	—
229	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
230	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
231	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
232	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
233	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
234	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
235	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
236	C₂²×D₆₇	—	4	1	no	—
237	C₂²×D₆₇	—	4	1	no	—
238	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
239	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
240	C₂²×D₆₇	#125	1	2	yes	—
241	C₂²×D₆₇	C₄⋊Q₈	1	2	yes	—
242	C₂×C₄×C₆₇	—	4	1	no	—
243	C₂×C₄×C₆₇	—	4	1	no	—
244	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
245	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
246	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
247	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
248	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
249	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
252	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
254	C₂×Dic₆₇	—	4	1	no	—
255	C₂×Dic₆₇	—	4	1	no	—
256	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
257	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
258	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
259	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
260	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
261	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
262	C₂×Dic₆₇	#125	1	2	yes	—
263	C₂×Dic₆₇	C₄⋊Q₈	1	2	yes	—
264	C₄×D₆₇	C₄×D₄	1	2	yes	—
265	C₄×D₆₇	C₄×D₄	1	2	yes	—
266	C₄×D₆₇	C₄×Q₈	1	2	yes	—
267	C₄×D₆₇	C₄×Q₈	1	2	yes	—
268	C₄×D₆₇	#121	1	2	yes	—
269	C₄×D₆₇	#121	1	2	yes	—
270	C₄×D₆₇	C₂²⋊Q₈	1	2	yes	—
271	C₄×D₆₇	C₂²⋊Q₈	1	2	yes	—
272	C₈×C₆₇	—	2	2	no	—
273	C₈×C₆₇	—	2	2	no	—
274	C₈×C₆₇	—	2	2	no	—
275	C₈×C₆₇	—	2	2	no	—
276	C₈×C₆₇	—	4	1	no	—
277	C₈×C₆₇	—	4	1	no	—
278	C₆₇⋊C₈	—	2	2	no	—
279	C₆₇⋊C₈	—	2	2	no	—
280	C₆₇⋊C₈	—	2	2	no	—
281	C₆₇⋊C₈	—	2	2	no	—
282	C₆₇⋊C₈	—	4	1	no	—
283	C₆₇⋊C₈	—	4	1	no	—
284	C₂²×C₆₇	—	2	1	no	—
285	C₂²×C₆₇	—	2	1	no	—
286	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
287	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
288	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
289	C₂×D₆₇	—	2	1	no	—
290	C₂×D₆₇	—	2	1	no	—
291	C₂×D₆₇	—	2	1	no	—
292	C₂×D₆₇	—	2	1	no	—
293	C₂×D₆₇	—	4	2	no	—
294	C₂×D₆₇	—	4	2	no	—
295	C₂×D₆₇	#389	1	2	yes	—
296	C₂×D₆₇	#389	1	2	yes	—
297	C₄×C₆₇	—	2	1	no	—
298	C₄×C₆₇	—	2	1	no	—
299	C₄×C₆₇	—	2	1	no	—
300	C₄×C₆₇	—	2	1	no	—
301	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
302	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
303	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
304	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
305	Dic₆₇	—	2	1	no	—
306	Dic₆₇	—	2	1	no	—
307	Dic₆₇	—	2	1	no	—
308	Dic₆₇	—	2	1	no	—
309	Dic₆₇	#389	1	2	yes	—
310	Dic₆₇	#389	1	2	yes	—
311	#21178	—	67	1	no	maximal, Sylow
312	C₂×C₆₇	—	2	1	no	—
313	C₂×C₆₇	—	2	1	no	—
314	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
315	C₂×C₆₇	#2916	1	1	yes	characteristic
316	C₂×C₆₇	#2916	1	1	yes	characteristic
317	D₆₇	—	2	2	no	—
318	D₆₇	—	2	2	no	—
319	C₂×#430	—	67	1	no	—
320	C₂²×#108	—	67	1	no	—
321	#2905	—	67	1	no	—
322	C₂×#424	—	67	1	no	—
323	#2873	—	67	1	no	—
324	#1950	—	67	1	no	—
325	#1950	—	67	1	no	—
326	#1950	—	67	1	no	—
327	#1950	—	67	1	no	—
328	#1950	—	67	1	no	—
329	#1950	—	67	1	no	—
330	#1950	—	67	1	no	—
331	#1950	D₆₇	1	1	yes	characteristic, complemented
332	#1738	—	67	1	no	—
333	C₂×#341	—	67	1	no	—
334	C₆₇	#21178	1	1	yes	characteristic, complemented, Sylow
335	C₂²×M₁₆	—	67	1	no	—
336	C₂²×M₁₆	—	67	1	no	—
337	C₂²×M₁₆	—	67	1	no	—
338	C₂²×M₁₆	—	67	1	no	—
339	C₂²×M₁₆	—	67	1	no	—
340	C₂³×C₈	—	67	1	no	—
341	C₂×#117	—	67	1	no	—
342	#430	—	134	4	no	—
343	C₂×#108	—	67	1	no	—
344	C₂×#108	—	67	1	no	—
345	C₂×#108	—	67	1	no	—
346	C₂×#108	—	67	1	no	—
347	C₂×#108	—	67	1	no	—
348	C₂×#108	—	67	1	no	—
349	C₂×#108	—	134	4	no	—
350	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
351	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
352	#424	—	67	2	no	—
353	#424	—	67	2	no	—
354	#424	—	67	2	no	—
355	#424	—	67	2	no	—
356	#424	—	134	1	no	—
357	#424	—	134	1	no	—
358	#424	—	134	1	no	—
359	#424	—	134	1	no	—
360	#423	—	67	1	no	—
361	#423	—	67	1	no	—
362	#423	—	67	1	no	—
363	#423	C₂×D₆₇	1	1	yes	characteristic
364	C₂×C₄⋊C₈	—	67	1	no	—
365	C₂×C₄⋊C₈	—	67	1	no	—
366	C₂×C₄⋊C₈	—	67	1	no	—
367	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
368	#405	—	134	1	no	—
369	#405	—	134	1	no	—
370	#405	—	134	1	no	—
371	#405	—	134	1	no	—
372	C₄×M₁₆	—	67	1	no	—
373	C₄×M₁₆	—	67	1	no	—
374	C₄×M₁₆	—	67	1	no	—
375	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
376	#341	—	67	1	no	—
377	#341	—	67	1	no	—
378	#341	—	67	1	no	—
379	#341	—	67	1	no	—
380	#341	—	67	1	no	—
381	#341	—	67	1	no	—
382	#341	C₂×D₆₇	1	1	yes	characteristic
383	#341	C₂×D₆₇	1	1	yes	characteristic
384	C₂³×C₄	—	67	1	no	—
385	C₂×M₁₆	—	67	2	no	—
386	C₂×M₁₆	—	67	2	no	—
387	C₂×M₁₆	—	67	1	no	—
388	C₂×M₁₆	—	67	1	no	—
389	C₂×M₁₆	—	67	4	no	—
390	C₂×M₁₆	—	67	4	no	—
391	C₂×M₁₆	—	67	1	no	—
392	C₂×M₁₆	—	67	1	no	—
393	C₂×M₁₆	—	134	1	no	—
394	C₂×M₁₆	—	134	1	no	—
395	C₂×M₁₆	—	134	2	no	—
396	C₂×M₁₆	—	134	1	no	—
397	C₂×M₁₆	—	134	1	no	—
398	C₂×M₁₆	—	134	2	no	—
399	C₂×M₁₆	—	134	2	no	—
400	C₂×M₁₆	—	134	2	no	—
401	C₂×M₁₆	—	134	2	no	—
402	C₂×M₁₆	—	134	2	no	—
403	C₂×M₁₆	—	134	2	no	—
404	C₂×M₁₆	—	134	2	no	—
405	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
406	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
407	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
408	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
409	C₂²×C₈	—	67	1	no	—
410	C₂²×C₈	—	67	1	no	—
411	C₂²×C₈	—	67	2	no	—
412	C₂²×C₈	—	67	2	no	—
413	C₂²×C₈	—	134	1	no	—
414	C₂²×C₈	—	134	1	no	—
415	C₂²×C₈	—	134	2	no	—
416	C₂²×C₈	—	134	2	no	—
417	C₂²×C₈	—	134	1	no	—
418	C₂²×C₈	—	134	1	no	—
419	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
420	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
421	#117	—	67	1	no	—
422	#117	—	67	1	no	—
423	#117	—	67	1	no	—
424	#117	—	67	1	no	—
425	#117	—	67	1	no	—
426	#117	—	67	1	no	—
427	#117	—	67	1	no	—
428	#117	—	67	1	no	—
429	C₂×C₄⋊C₄	—	67	1	no	—
430	C₂×C₄⋊C₄	—	67	1	no	—
431	C₂×C₂²⋊C₄	—	67	1	no	—
432	C₂×C₂²⋊C₄	—	67	1	no	—
433	C₂×C₄²	—	67	1	no	—
434	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
435	#108	—	2	2	no	—
436	#108	—	2	2	no	—
437	#108	—	134	2	no	—
438	#108	—	134	2	no	—
439	#108	—	134	2	no	—
440	#108	—	134	2	no	—
441	#108	—	134	2	no	—
442	#108	—	134	2	no	—
443	C₄⋊C₈	—	2	1	no	—
444	C₄⋊C₈	—	2	1	no	—
445	C₄⋊C₈	—	67	2	no	—
446	C₄⋊C₈	—	67	2	no	—
447	C₄⋊C₈	—	67	2	no	—
448	C₄⋊C₈	—	67	2	no	—
449	C₄⋊C₈	—	67	2	no	—
450	C₄⋊C₈	—	67	2	no	—
451	C₄⋊C₈	—	134	1	no	—
452	C₄⋊C₈	—	134	1	no	—
453	C₄⋊C₈	—	134	1	no	—
454	C₄⋊C₈	—	134	1	no	—
455	C₄⋊C₈	—	134	1	no	—
456	C₄⋊C₈	—	134	1	no	—
457	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
458	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
459	#97	—	2	1	no	—
460	#97	—	134	1	no	—
461	#97	—	134	1	no	—
462	#97	—	134	1	no	—
463	C₄×C₈	—	2	1	no	—
464	C₄×C₈	—	134	1	no	—
465	C₄×C₈	—	134	1	no	—
466	C₄×C₈	—	134	1	no	—
467	C₂⁴	—	67	1	no	—
468	C₂²×C₄	—	67	1	no	—
469	C₂²×C₄	—	67	1	no	—
470	C₂²×C₄	—	67	1	no	—
471	C₂²×C₄	—	67	1	no	—
472	C₂²×C₄	—	67	1	no	—
473	C₂²×C₄	—	67	2	no	—
474	C₂²×C₄	—	67	1	no	—
475	C₂²×C₄	—	67	1	no	—
476	C₂²×C₄	—	67	2	no	—
477	C₂²×C₄	—	67	2	no	—
478	C₂²×C₄	—	67	2	no	—
479	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
480	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
481	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
482	M₁₆	—	2	1	no	—
483	M₁₆	—	2	1	no	—
484	M₁₆	—	2	1	no	—
485	M₁₆	—	2	1	no	—
486	M₁₆	—	2	1	no	—
487	M₁₆	—	2	1	no	—
488	M₁₆	—	268	2	no	—
489	M₁₆	—	268	2	no	—
490	M₁₆	—	67	2	no	—
491	M₁₆	—	67	2	no	—
492	M₁₆	—	134	4	no	—
493	M₁₆	—	134	4	no	—
494	M₁₆	—	134	4	no	—
495	M₁₆	—	134	4	no	—
496	M₁₆	—	134	1	no	—
497	M₁₆	—	134	1	no	—
498	M₁₆	—	134	1	no	—
499	M₁₆	—	134	1	no	—
500	M₁₆	—	134	1	no	—
501	M₁₆	—	134	1	no	—
502	M₁₆	D₄×D₆₇	1	2	yes	—
503	M₁₆	Q₈×D₆₇	1	2	yes	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	268	2	no	—
513	C₂×C₈	—	268	2	no	—
514	C₂×C₈	—	67	2	no	—
515	C₂×C₈	—	67	2	no	—
516	C₂×C₈	—	134	4	no	—
517	C₂×C₈	—	134	4	no	—
518	C₂×C₈	—	134	4	no	—
519	C₂×C₈	—	134	4	no	—
520	C₂×C₈	—	134	1	no	—
521	C₂×C₈	—	134	1	no	—
522	C₂×C₈	—	134	1	no	—
523	C₂×C₈	—	134	1	no	—
524	C₂×C₈	—	134	1	no	—
525	C₂×C₈	—	134	1	no	—
526	C₂×C₈	—	134	1	no	—
527	C₂×C₈	—	134	1	no	—
528	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
529	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
530	C₄⋊C₄	—	67	1	no	—
531	C₄⋊C₄	—	67	1	no	—
532	C₄⋊C₄	—	67	1	no	—
533	C₄⋊C₄	—	67	1	no	—
534	C₄⋊C₄	—	67	1	no	—
535	C₄⋊C₄	—	67	1	no	—
536	C₄⋊C₄	—	67	1	no	—
537	C₄⋊C₄	—	67	1	no	—
538	C₂²⋊C₄	—	67	1	no	—
539	C₂²⋊C₄	—	67	1	no	—
540	C₂²⋊C₄	—	67	1	no	—
541	C₂²⋊C₄	—	67	1	no	—
542	C₂²⋊C₄	—	67	1	no	—
543	C₂²⋊C₄	—	67	1	no	—
544	C₂²⋊C₄	—	67	1	no	—
545	C₂²⋊C₄	—	67	1	no	—
546	C₄²	—	67	1	no	—
547	C₄²	—	67	1	no	—
548	C₄²	—	67	1	no	—
549	C₄²	—	67	1	no	—
550	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
551	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
552	C₄²	D₄×D₆₇	1	1	yes	characteristic
553	C₄²	D₄×D₆₇	1	1	yes	characteristic
554	C₂³	—	268	1	no	—
555	C₂³	—	268	1	no	—
556	C₂³	—	67	2	no	—
557	C₂³	—	67	2	no	—
558	C₂³	—	67	1	no	—
559	C₂³	—	67	1	no	—
560	C₂³	C₂×Q₈×D₆₇	1	1	yes	characteristic
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	268	1	no	—
564	C₂×C₄	—	268	1	no	—
565	C₂×C₄	—	67	1	no	—
566	C₂×C₄	—	67	1	no	—
567	C₂×C₄	—	67	2	no	—
568	C₂×C₄	—	67	2	no	—
569	C₂×C₄	—	67	2	no	—
570	C₂×C₄	—	67	2	no	—
571	C₂×C₄	—	67	2	no	—
572	C₂×C₄	—	67	2	no	—
573	C₂×C₄	—	67	2	no	—
574	C₂×C₄	—	67	2	no	—
575	C₂×C₄	—	67	2	no	—
576	C₂×C₄	—	67	2	no	—
577	C₂×C₄	—	67	1	no	—
578	C₂×C₄	—	67	1	no	—
579	C₂×C₄	—	67	1	no	—
580	C₂×C₄	—	67	1	no	—
581	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
582	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
583	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
584	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
585	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
586	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
587	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
588	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
589	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
590	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
591	C₈	—	2	2	no	—
592	C₈	—	2	2	no	—
593	C₈	—	2	2	no	—
594	C₈	—	2	2	no	—
595	C₈	—	4	1	no	—
596	C₈	—	4	1	no	—
597	C₈	—	268	1	no	—
598	C₈	—	268	1	no	—
599	C₈	—	134	2	no	—
600	C₈	—	134	2	no	—
601	C₈	—	134	2	no	—
602	C₈	—	134	2	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	268	2	no	—
606	C₂²	—	268	2	no	—
607	C₂²	—	67	2	no	—
608	C₂²	—	67	2	no	—
609	C₂²	—	134	1	no	—
610	C₂²	—	134	1	no	—
611	C₂²	—	134	1	no	—
612	C₂²	—	134	1	no	—
613	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
614	C₂²	#125×D₆₇	1	1	yes	characteristic
615	C₂²	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
616	C₄	—	2	1	no	—
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	2	1	no	—
620	C₄	—	67	2	no	—
621	C₄	—	67	2	no	—
622	C₄	—	134	1	no	—
623	C₄	—	134	1	no	—
624	C₄	—	134	1	no	—
625	C₄	—	134	1	no	—
626	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
627	C₄	C₄×Q₈×D₆₇	1	1	yes	characteristic
628	C₄	#121×D₆₇	1	1	yes	characteristic
629	C₄	C₂²⋊Q₈×D₆₇	1	1	yes	characteristic
630	C₂	—	2	1	no	—
631	C₂	—	2	1	no	—
632	C₂	—	134	2	no	—
633	C₂	—	134	2	no	—
634	C₂	#389×D₆₇	1	1	yes	characteristic
635	C₂	#22568817	1	1	yes	characteristic
636	C₂	#22568819	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1950	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1950	D₆₇
hypocenter	C₆₇	#21178
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
6. #22560985	C₂	semidirect
7. #22560984	C₂	semidirect
8. #22560983	C₂	semidirect
9. #22560983	C₂	semidirect
10. #22560982	C₂	semidirect
11. #22560981	C₂	semidirect
12. #22560981	C₂	semidirect
15. C₆₇×#1950	C₂	semidirect
331. #1950	D₆₇	semidirect
334. C₆₇	#21178	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16863443	2
C₂×#108×D₆₇	2
#22568771	2
C₂×#16863421	2
#22568626	2
#22560985	2
#22560984	2
#22560983	2
#22560982	2
#22560981	2
#22559934	2
#341×D₆₇	2
C₆₇×#1950	2
#21178	67