Subgroups — #30761690

Order 17152.

794 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863419	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863161	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863160	C₂	1	1	yes	maximal, characteristic
6	C₂×#16863160	C₂	1	1	yes	maximal, characteristic
7	C₂×#341×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11761612	C₂²	1	1	yes	characteristic
9	C₂²×#11761612	C₂²	1	1	yes	characteristic
10	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
12	#16863419	—	2	2	no	—
13	#16863419	C₄	1	4	yes	—
14	C₂×Dic₆₇⋊C₈	—	2	2	no	—
15	C₂×Dic₆₇⋊C₈	—	2	2	no	—
16	C₂×Dic₆₇⋊C₈	—	2	1	no	—
17	C₂×Dic₆₇⋊C₈	—	2	1	no	—
18	C₂×Dic₆₇⋊C₈	C₄	1	2	yes	—
19	C₂×Dic₆₇⋊C₈	C₄	1	2	yes	—
20	M₁₆×Dic₆₇	—	2	2	no	—
21	M₁₆×Dic₆₇	C₄	1	4	yes	—
22	C₂×#14523953	—	2	1	no	—
23	C₂×C₈×Dic₆₇	—	2	1	no	—
24	#16863161	—	2	2	no	—
25	#16863160	—	2	2	no	—
26	#16863160	—	2	2	no	—
27	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
29	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	#341×C₆₇	—	2	2	no	—
31	C₂³×Dic₆₇	Q₈	1	1	yes	characteristic
32	C₂³×Dic₆₇	Q₈	1	1	yes	characteristic
33	C₂×#11761612	—	2	1	no	—
34	C₂×#11761612	—	2	1	no	—
35	C₂×#11761612	—	2	1	no	—
36	C₂×#11761612	—	2	1	no	—
37	C₂×#11761612	—	2	1	no	—
38	C₂×#11761612	—	2	1	no	—
39	C₂×#11761612	—	2	1	no	—
40	C₂×#11761612	—	2	1	no	—
41	C₂×#11761612	C₂×C₄	1	2	yes	—
42	C₂×#11761612	C₂×C₄	1	2	yes	—
43	C₂×#11761612	D₄	1	2	yes	—
44	C₂×#11761612	Q₈	1	2	yes	—
45	C₂²×C₆₇⋊C₈	—	2	1	no	—
46	C₂²×C₆₇⋊C₈	—	2	1	no	—
47	C₂²×C₆₇⋊C₈	—	2	1	no	—
48	C₂²×C₆₇⋊C₈	—	2	1	no	—
49	C₂²×C₆₇⋊C₈	—	2	1	no	—
50	C₂²×C₆₇⋊C₈	—	2	1	no	—
51	C₂²×C₆₇⋊C₈	—	2	1	no	—
52	C₂²×C₆₇⋊C₈	—	2	1	no	—
53	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
54	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
59	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
60	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
61	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
62	Dic₆₇⋊C₈	—	2	4	no	—
63	Dic₆₇⋊C₈	—	2	4	no	—
64	Dic₆₇⋊C₈	—	2	4	no	—
65	Dic₆₇⋊C₈	—	2	4	no	—
66	#14523953	—	2	4	no	—
67	C₈×Dic₆₇	—	2	4	no	—
68	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×M₁₆×C₆₇	—	2	1	no	—
70	C₂×M₁₆×C₆₇	—	2	1	no	—
71	C₂×M₁₆×C₆₇	—	2	1	no	—
72	C₂×M₁₆×C₆₇	—	2	1	no	—
73	C₂×M₁₆×C₆₇	—	2	1	no	—
74	C₂×M₁₆×C₆₇	—	2	1	no	—
75	C₂×M₁₆×C₆₇	—	2	1	no	—
76	C₂×M₁₆×C₆₇	—	2	1	no	—
77	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
78	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
79	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
80	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
81	C₂²×C₈×C₆₇	—	2	1	no	—
82	C₂²×C₈×C₆₇	—	2	1	no	—
83	C₂²×C₈×C₆₇	—	2	1	no	—
84	C₂²×C₈×C₆₇	—	2	1	no	—
85	C₂²×C₈×C₆₇	—	2	1	no	—
86	C₂²×C₈×C₆₇	—	2	1	no	—
87	C₂²×C₈×C₆₇	—	2	1	no	—
88	C₂²×C₈×C₆₇	—	2	1	no	—
89	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
90	C₂²×Dic₆₇	—	2	4	no	—
91	C₂²×Dic₆₇	—	4	2	no	—
92	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
94	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
95	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
96	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
97	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
98	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
99	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
100	#11761612	—	2	2	no	—
101	#11761612	—	2	2	no	—
102	#11761612	—	2	2	no	—
103	#11761612	—	2	2	no	—
104	#11761612	—	2	2	no	—
105	#11761612	—	4	1	no	—
106	#11761612	—	4	1	no	—
107	#11761612	C₄⋊C₄	1	4	yes	—
108	C₂×C₆₇⋊C₈	—	2	2	no	—
109	C₂×C₆₇⋊C₈	—	2	2	no	—
110	C₂×C₆₇⋊C₈	—	2	2	no	—
111	C₂×C₆₇⋊C₈	—	2	2	no	—
112	C₂×C₆₇⋊C₈	—	2	2	no	—
113	C₂×C₆₇⋊C₈	—	2	2	no	—
114	C₂×C₆₇⋊C₈	—	2	2	no	—
115	C₂×C₆₇⋊C₈	—	4	1	no	—
116	C₂×C₆₇⋊C₈	—	4	1	no	—
117	C₂×C₆₇⋊C₈	—	4	1	no	—
118	C₂×C₆₇⋊C₈	—	4	1	no	—
119	C₂×C₆₇⋊C₈	—	4	1	no	—
120	C₂×C₆₇⋊C₈	—	4	1	no	—
121	C₄×Dic₆₇	C₄²	1	2	yes	—
122	C₄×Dic₆₇	C₄²	1	2	yes	—
123	C₄×Dic₆₇	C₂²⋊C₄	1	2	yes	—
124	C₄×Dic₆₇	C₂²⋊C₄	1	2	yes	—
125	C₄×Dic₆₇	C₂²⋊C₄	1	2	yes	—
126	C₄×Dic₆₇	C₂²⋊C₄	1	2	yes	—
127	C₄×Dic₆₇	C₂²⋊C₄	1	2	yes	—
128	C₄×Dic₆₇	C₂²⋊C₄	1	2	yes	—
129	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
144	M₁₆×C₆₇	—	2	2	no	—
145	M₁₆×C₆₇	—	2	2	no	—
146	M₁₆×C₆₇	—	2	2	no	—
147	M₁₆×C₆₇	—	2	2	no	—
148	M₁₆×C₆₇	—	2	2	no	—
149	M₁₆×C₆₇	—	4	1	no	—
150	M₁₆×C₆₇	—	4	1	no	—
151	M₁₆×C₆₇	C₄⋊C₄	1	4	yes	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	2	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	4	1	no	—
160	C₂×C₈×C₆₇	—	4	1	no	—
161	C₂×C₈×C₆₇	—	4	1	no	—
162	C₂×C₈×C₆₇	—	4	1	no	—
163	C₂×C₈×C₆₇	—	4	1	no	—
164	C₂×C₈×C₆₇	—	4	1	no	—
165	C₂³×C₆₇	—	2	2	no	—
166	C₂³×C₆₇	—	2	2	no	—
167	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂³×C₆₇	#125	1	1	yes	characteristic
171	C₂³×C₆₇	#125	1	1	yes	characteristic
172	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
173	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
181	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
190	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
191	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
192	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
193	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
198	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
199	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
200	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
202	C₂×Dic₆₇	—	2	2	no	—
203	C₂×Dic₆₇	—	2	2	no	—
204	C₂×Dic₆₇	—	2	2	no	—
205	C₂×Dic₆₇	—	2	4	no	—
206	C₂×Dic₆₇	—	2	4	no	—
207	C₂×Dic₆₇	—	4	2	no	—
208	C₂×Dic₆₇	—	4	2	no	—
209	C₂×Dic₆₇	#95	1	2	yes	—
210	C₂×Dic₆₇	#95	1	2	yes	—
211	C₂×Dic₆₇	#95	1	2	yes	—
212	C₂×Dic₆₇	#95	1	2	yes	—
213	C₂×Dic₆₇	#108	1	4	yes	—
214	C₈×C₆₇	—	2	4	no	—
215	C₈×C₆₇	—	4	2	no	—
216	C₈×C₆₇	—	4	2	no	—
217	C₆₇⋊C₈	—	2	4	no	—
218	C₆₇⋊C₈	—	4	2	no	—
219	C₆₇⋊C₈	—	4	2	no	—
220	C₂²×C₆₇	—	2	2	no	—
221	C₂²×C₆₇	—	2	2	no	—
222	C₂²×C₆₇	—	2	2	no	—
223	C₂²×C₆₇	—	2	2	no	—
224	C₂²×C₆₇	—	2	1	no	—
225	C₂²×C₆₇	—	2	1	no	—
226	C₂²×C₆₇	—	2	1	no	—
227	C₂²×C₆₇	—	2	1	no	—
228	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
229	C₂²×C₆₇	#382	1	1	yes	characteristic
230	C₂²×C₆₇	#384	1	1	yes	characteristic
231	C₂²×C₆₇	#389	1	1	yes	characteristic
232	C₂²×C₆₇	#389	1	1	yes	characteristic
233	C₂²×C₆₇	#389	1	1	yes	characteristic
234	C₂²×C₆₇	#389	1	1	yes	characteristic
235	C₂²×C₆₇	C₂×#108	1	2	yes	—
236	C₂²×C₆₇	#430	1	2	yes	—
237	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
238	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
239	C₄×C₆₇	#380	1	1	yes	characteristic
240	C₄×C₆₇	#380	1	1	yes	characteristic
241	C₄×C₆₇	#390	1	1	yes	characteristic
242	C₄×C₆₇	#390	1	1	yes	characteristic
243	C₄×C₆₇	#391	1	1	yes	characteristic
244	C₄×C₆₇	#391	1	1	yes	characteristic
245	Dic₆₇	—	2	2	no	—
246	Dic₆₇	—	2	2	no	—
247	Dic₆₇	—	2	2	no	—
248	Dic₆₇	#341	1	4	yes	complemented
249	#10853	—	67	1	no	maximal, Sylow
250	C₂×C₆₇	—	2	1	no	—
251	C₂×C₆₇	—	2	1	no	—
252	C₂×C₆₇	—	2	1	no	—
253	C₂×C₆₇	—	2	1	no	—
254	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
255	C₂×C₆₇	C₂×#341	1	1	yes	characteristic
256	C₂×C₆₇	#1738	1	1	yes	characteristic
257	C₂×C₆₇	#1950	1	2	yes	—
258	C₂×C₆₇	#1950	1	2	yes	—
259	C₂×#423	—	67	1	no	—
260	C₂²×C₄⋊C₈	—	67	1	no	—
261	C₂×C₄×M₁₆	—	67	1	no	—
262	C₂×#341	—	67	1	no	—
263	C₂×#341	—	67	1	no	—
264	C₂×#341	—	67	1	no	—
265	C₂×#341	D₆₇	1	1	yes	characteristic
266	C₆₇	#10853	1	1	yes	characteristic, complemented, Sylow
267	C₂²×M₁₆	—	67	1	no	—
268	C₂²×M₁₆	—	67	1	no	—
269	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
270	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
271	C₂³×C₈	—	67	1	no	—
272	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
273	C₂²×C₄²	—	67	1	no	—
274	#423	—	67	4	no	—
275	#423	—	134	2	no	—
276	C₂×C₄⋊C₈	—	67	2	no	—
277	C₂×C₄⋊C₈	—	67	2	no	—
278	C₂×C₄⋊C₈	—	134	2	no	—
279	C₂×C₄⋊C₈	—	134	2	no	—
280	C₂×C₄⋊C₈	—	134	1	no	—
281	C₂×C₄⋊C₈	—	134	1	no	—
282	C₄×M₁₆	—	67	4	no	—
283	C₄×M₁₆	—	134	2	no	—
284	C₂×#97	—	134	1	no	—
285	C₂×C₄×C₈	—	134	1	no	—
286	#341	—	2	2	no	—
287	#341	—	134	2	no	—
288	#341	—	134	2	no	—
289	#341	—	134	2	no	—
290	C₂³×C₄	—	67	1	no	—
291	C₂³×C₄	—	67	1	no	—
292	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
293	C₂×M₁₆	—	2	1	no	—
294	C₂×M₁₆	—	2	1	no	—
295	C₂×M₁₆	—	2	1	no	—
296	C₂×M₁₆	—	2	1	no	—
297	C₂×M₁₆	—	2	1	no	—
298	C₂×M₁₆	—	2	1	no	—
299	C₂×M₁₆	—	2	1	no	—
300	C₂×M₁₆	—	2	1	no	—
301	C₂×M₁₆	—	67	2	no	—
302	C₂×M₁₆	—	67	2	no	—
303	C₂×M₁₆	—	67	2	no	—
304	C₂×M₁₆	—	67	2	no	—
305	C₂×M₁₆	—	134	1	no	—
306	C₂×M₁₆	—	134	1	no	—
307	C₂×M₁₆	—	134	1	no	—
308	C₂×M₁₆	—	134	1	no	—
309	C₂×M₁₆	—	134	1	no	—
310	C₂×M₁₆	—	134	1	no	—
311	C₂×M₁₆	—	134	1	no	—
312	C₂×M₁₆	—	134	1	no	—
313	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
314	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
315	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
316	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
317	C₂²×C₈	—	2	1	no	—
318	C₂²×C₈	—	2	1	no	—
319	C₂²×C₈	—	2	1	no	—
320	C₂²×C₈	—	2	1	no	—
321	C₂²×C₈	—	2	1	no	—
322	C₂²×C₈	—	2	1	no	—
323	C₂²×C₈	—	2	1	no	—
324	C₂²×C₈	—	2	1	no	—
325	C₂²×C₈	—	67	2	no	—
326	C₂²×C₈	—	134	1	no	—
327	C₂²×C₈	—	134	1	no	—
328	C₂²×C₈	—	134	1	no	—
329	C₂²×C₈	—	134	1	no	—
330	C₂²×C₈	—	134	1	no	—
331	C₂²×C₈	—	134	1	no	—
332	C₂²×C₈	—	134	1	no	—
333	C₂²×C₈	—	134	1	no	—
334	C₂²×C₈	C₄×D₆₇	1	2	yes	—
335	C₂×C₄²	—	67	2	no	—
336	C₂×C₄²	—	67	1	no	—
337	C₂×C₄²	—	67	1	no	—
338	C₂×C₄²	—	67	2	no	—
339	C₂×C₄²	—	67	2	no	—
340	C₂×C₄²	—	67	2	no	—
341	C₂×C₄²	—	67	1	no	—
342	C₂×C₄²	—	67	1	no	—
343	C₄⋊C₈	—	134	4	no	—
344	C₄⋊C₈	—	134	4	no	—
345	C₄⋊C₈	—	134	4	no	—
346	C₄⋊C₈	—	134	4	no	—
347	#97	—	134	4	no	—
348	C₄×C₈	—	134	4	no	—
349	C₂⁴	Q₈×D₆₇	1	1	yes	characteristic
350	C₂²×C₄	—	268	2	no	—
351	C₂²×C₄	—	67	2	no	—
352	C₂²×C₄	—	67	2	no	—
353	C₂²×C₄	—	67	1	no	—
354	C₂²×C₄	—	67	2	no	—
355	C₂²×C₄	—	67	2	no	—
356	C₂²×C₄	—	67	1	no	—
357	C₂²×C₄	—	67	1	no	—
358	C₂²×C₄	—	67	1	no	—
359	C₂²×C₄	—	134	4	no	—
360	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
361	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
362	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
363	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
364	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
365	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
366	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
367	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
368	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
369	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
370	C₂²×C₄	#11761627	1	1	yes	characteristic
371	C₂²×C₄	#11761627	1	1	yes	characteristic
372	C₂²×C₄	#11761628	1	1	yes	characteristic
373	C₂²×C₄	#11761628	1	1	yes	characteristic
374	M₁₆	—	2	2	no	—
375	M₁₆	—	2	2	no	—
376	M₁₆	—	2	2	no	—
377	M₁₆	—	2	2	no	—
378	M₁₆	—	2	2	no	—
379	M₁₆	—	4	1	no	—
380	M₁₆	—	4	1	no	—
381	M₁₆	—	268	1	no	—
382	M₁₆	—	268	1	no	—
383	M₁₆	—	67	4	no	—
384	M₁₆	—	134	2	no	—
385	M₁₆	—	134	2	no	—
386	M₁₆	—	134	2	no	—
387	M₁₆	—	134	2	no	—
388	M₁₆	—	134	2	no	—
389	M₁₆	Dic₆₇⋊C₄	1	4	yes	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	4	1	no	—
398	C₂×C₈	—	4	1	no	—
399	C₂×C₈	—	4	1	no	—
400	C₂×C₈	—	4	1	no	—
401	C₂×C₈	—	4	1	no	—
402	C₂×C₈	—	4	1	no	—
403	C₂×C₈	—	268	1	no	—
404	C₂×C₈	—	268	1	no	—
405	C₂×C₈	—	268	1	no	—
406	C₂×C₈	—	268	1	no	—
407	C₂×C₈	—	268	1	no	—
408	C₂×C₈	—	268	1	no	—
409	C₂×C₈	—	134	2	no	—
410	C₂×C₈	—	134	2	no	—
411	C₂×C₈	—	134	2	no	—
412	C₂×C₈	—	134	2	no	—
413	C₂×C₈	—	134	2	no	—
414	C₂×C₈	—	134	2	no	—
415	C₂×C₈	—	134	2	no	—
416	C₄²	—	67	2	no	—
417	C₄²	—	67	2	no	—
418	C₄²	—	67	2	no	—
419	C₄²	—	67	2	no	—
420	C₄²	—	67	2	no	—
421	C₄²	—	67	2	no	—
422	C₄²	—	67	2	no	—
423	C₄²	—	67	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	2	no	—
426	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
427	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
428	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
429	C₂³	#14524033	1	1	yes	characteristic
430	C₂³	#14524033	1	1	yes	characteristic
431	C₂³	#14524041	1	1	yes	characteristic
432	C₂³	#14524042	1	1	yes	characteristic
433	C₂×C₄	—	268	2	no	—
434	C₂×C₄	—	268	2	no	—
435	C₂×C₄	—	67	2	no	—
436	C₂×C₄	—	67	4	no	—
437	C₂×C₄	—	67	2	no	—
438	C₂×C₄	—	67	2	no	—
439	C₂×C₄	—	67	2	no	—
440	C₂×C₄	—	134	4	no	—
441	C₂×C₄	—	134	4	no	—
442	C₂×C₄	—	134	2	no	—
443	C₂×C₄	—	134	2	no	—
444	C₂×C₄	—	134	2	no	—
445	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
446	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
447	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
448	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
449	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
450	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
451	C₂×C₄	#14523999	1	1	yes	characteristic
452	C₂×C₄	#14524000	1	1	yes	characteristic
453	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
454	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
455	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
456	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
457	C₂×C₄	#14524008	1	1	yes	characteristic
458	C₂×C₄	#14524008	1	1	yes	characteristic
459	C₂×C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
460	C₂×C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
461	C₂×C₄	#14524015	1	1	yes	characteristic
462	C₂×C₄	#14524015	1	1	yes	characteristic
463	C₂×C₄	#14524016	1	1	yes	characteristic
464	C₂×C₄	#14524016	1	1	yes	characteristic
465	C₂×C₄	#14524020	1	1	yes	characteristic
466	C₂×C₄	#14524020	1	1	yes	characteristic
467	C₂×C₄	#14524021	1	1	yes	characteristic
468	C₂×C₄	#14524021	1	1	yes	characteristic
469	C₂×C₄	#14524029	1	1	yes	characteristic
470	C₂×C₄	#14524029	1	1	yes	characteristic
471	C₂×C₄	#14524039	1	1	yes	characteristic
472	C₂×C₄	#14524042	1	1	yes	characteristic
473	C₈	—	2	4	no	—
474	C₈	—	4	2	no	—
475	C₈	—	4	2	no	—
476	C₈	—	268	2	no	—
477	C₈	—	268	2	no	—
478	C₈	—	134	4	no	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	2	2	no	—
484	C₂²	—	2	2	no	—
485	C₂²	—	2	2	no	—
486	C₂²	—	2	2	no	—
487	C₂²	#95×D₆₇	1	1	yes	characteristic
488	C₂²	#16863260	1	1	yes	characteristic
489	C₂²	#16863265	1	1	yes	characteristic
490	C₂²	#16863291	1	1	yes	characteristic
491	C₂²	#16863291	1	1	yes	characteristic
492	C₂²	#16863293	1	1	yes	characteristic
493	C₂²	#16863293	1	1	yes	characteristic
494	C₂²	#108×D₆₇	1	2	yes	—
495	C₂²	#16863443	1	2	yes	—
496	C₄	—	67	4	no	—
497	C₄	—	134	2	no	—
498	C₄	—	134	2	no	—
499	C₄	—	134	2	no	—
500	C₄	#16863247	1	1	yes	characteristic
501	C₄	C₄×Dic₆₇⋊C₄	1	1	yes	characteristic
502	C₄	#16863254	1	1	yes	characteristic
503	C₄	#16863255	1	1	yes	characteristic
504	C₄	#16863296	1	1	yes	characteristic
505	C₄	#16863296	1	1	yes	characteristic
506	C₄	#16863300	1	1	yes	characteristic
507	C₄	#16863300	1	1	yes	characteristic
508	C₂	—	2	1	no	—
509	C₂	—	2	1	no	—
510	C₂	—	2	1	no	—
511	C₂	—	2	1	no	—
512	C₂	Dic₆₇⋊#95	1	1	yes	characteristic
513	C₂	#341×D₆₇	1	1	yes	characteristic
514	C₂	#22559934	1	1	yes	characteristic
515	C₂	#22560981	1	2	yes	—
516	C₂	#22560983	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#341×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#341	D₆₇
hypocenter	C₆₇	#10853
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
248. Dic₆₇	#341	semidirect
266. C₆₇	#10853	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863419	2
C₂²×Dic₆₇⋊C₈	2
C₂×M₁₆×Dic₆₇	2
C₂×#16863161	2
C₂×#16863160	2
C₂×#341×C₆₇	2
#10853	67