Subgroups — #30824864

Order 17152.

712 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568841	C₂	1	1	yes	maximal, characteristic
2	#432×D₆₇	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
4	#22558977	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558976	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558974	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558971	C₂	1	2	yes	maximal, complemented
8	#22558969	C₂	1	2	yes	maximal, complemented
9	#22558960	C₂	1	2	yes	maximal, complemented
10	#22558958	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1569	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₆₇	C₄	1	2	yes	—
13	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
14	#16863453	C₂²	1	1	yes	characteristic
15	#16863452	—	2	1	no	—
16	#16863452	C₂²	1	1	yes	characteristic
17	#16863452	C₂²	1	1	yes	characteristic
18	#16863452	C₂²	1	2	yes	—
19	#16863451	—	2	1	no	—
20	#16863450	—	2	1	no	—
21	#16863448	—	2	1	no	—
22	#16863448	C₂²	1	2	yes	—
23	#16863420	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×D₆₇	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×D₆₇	C₂²	1	1	yes	characteristic
26	#16863359	—	2	1	no	—
27	C₂²⋊C₈×D₆₇	—	2	1	no	—
28	C₄×#11761612	C₂²	1	1	yes	characteristic
29	#16863348	C₄	1	1	yes	characteristic
30	#16863348	C₄	1	1	yes	characteristic
31	M₁₆×Dic₆₇	—	2	1	no	—
32	M₁₆×Dic₆₇	C₄	1	2	yes	—
33	C₄×#11761611	C₂²	1	2	yes	—
34	#97×D₆₇	—	2	1	no	—
35	#97×D₆₇	C₄	1	1	yes	characteristic
36	#97×D₆₇	C₄	1	1	yes	characteristic
37	C₄×C₈×D₆₇	C₂²	1	2	yes	—
38	#16863149	C₂²	1	2	yes	complemented
39	#16863148	C₂²	1	2	yes	complemented
40	#16863148	C₂²	1	2	yes	complemented
41	#16863147	C₂²	1	2	yes	complemented
42	#16863146	C₂²	1	2	yes	complemented
43	#16863146	C₂²	1	2	yes	complemented
44	#432×C₆₇	C₂²	1	1	yes	characteristic
45	#423×C₆₇	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
47	#335×C₆₇	C₂²	1	2	yes	complemented
48	#334×C₆₇	C₂²	1	2	yes	complemented
49	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₄×D₆₇	Q₈	1	1	yes	characteristic
52	C₂×#11761612	—	2	1	no	—
53	C₂×#11761612	C₂×C₄	1	2	yes	—
54	C₂×#11761611	C₂×C₄	1	2	yes	—
55	C₂×#11761611	C₂×C₄	1	2	yes	—
56	M₁₆×D₆₇	—	2	8	no	—
57	C₂×C₈×D₆₇	—	2	1	no	—
58	C₂×C₈×D₆₇	—	2	1	no	—
59	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
63	C₂×C₄×Dic₆₇	Q₈	1	1	yes	characteristic
64	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
65	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
66	C₄²×D₆₇	D₄	1	1	yes	characteristic
67	C₄²×D₆₇	D₄	1	1	yes	characteristic
68	C₄²×D₆₇	Q₈	1	1	yes	characteristic
69	C₄²×D₆₇	Q₈	1	1	yes	characteristic
70	C₄²×D₆₇	C₂³	1	1	yes	characteristic
71	C₄²×D₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
73	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
74	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
75	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
76	Dic₆₇⋊C₈	—	2	1	no	—
77	Dic₆₇⋊C₈	—	2	1	no	—
78	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
79	(C₂×D₆₇)⋊C₈	—	2	1	no	—
80	(C₂×D₆₇)⋊C₈	—	2	1	no	—
81	(C₂×D₆₇)⋊C₈	—	2	2	no	—
82	(C₂×D₆₇)⋊C₈	—	2	2	no	—
83	#14523954	—	2	1	no	—
84	#14523954	C₂×C₄	1	1	yes	characteristic
85	#14523954	C₂×C₄	1	1	yes	characteristic
86	#14523953	—	2	1	no	—
87	#14523953	—	2	1	no	—
88	#14523953	—	2	2	no	—
89	#14523953	C₂×C₄	1	2	yes	—
90	#14523953	C₂×C₄	1	2	yes	—
91	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
92	C₈×Dic₆₇	—	2	2	no	—
93	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
94	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
95	C₂×M₁₆×C₆₇	—	2	1	no	—
96	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₆₇	—	2	1	no	—
99	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
101	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
102	C₂²⋊C₈×C₆₇	—	2	1	no	—
103	#97×C₆₇	—	2	1	no	—
104	#97×C₆₇	C₂×C₄	1	1	yes	characteristic
105	#97×C₆₇	C₂×C₄	1	1	yes	characteristic
106	C₄×C₈×C₆₇	C₂³	1	2	yes	—
107	C₂³×D₆₇	C₄⋊C₄	1	1	yes	characteristic
108	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
109	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
110	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×D₆₇	—	2	4	no	—
112	C₂×C₄×D₆₇	—	4	1	no	—
113	C₂×C₄×D₆₇	—	4	1	no	—
114	C₂×C₄×D₆₇	—	4	2	no	—
115	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
126	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
127	#11761612	—	2	4	no	—
128	#11761612	—	4	2	no	—
129	#11761611	—	2	4	no	—
130	#11761611	M₁₆	1	8	yes	complemented
131	C₂×C₆₇⋊C₈	—	2	2	no	—
132	C₂×C₆₇⋊C₈	—	2	1	no	—
133	C₂×C₆₇⋊C₈	—	2	1	no	—
134	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
135	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
136	C₈×D₆₇	—	2	4	no	—
137	C₈×D₆₇	—	2	4	no	—
138	C₈×D₆₇	—	4	2	no	—
139	C₈×D₆₇	—	4	2	no	—
140	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
141	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
142	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
143	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
144	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
145	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
146	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
149	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
150	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
151	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
155	M₁₆×C₆₇	—	2	4	no	—
156	M₁₆×C₆₇	—	4	2	no	—
157	C₂×C₈×C₆₇	—	2	2	no	—
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	—	2	1	no	—
160	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
161	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
162	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
163	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
165	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
166	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂²×D₆₇	—	2	2	no	—
168	C₂²×D₆₇	—	2	2	no	—
169	C₂²×D₆₇	—	4	1	no	—
170	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
182	C₂×Dic₆₇	—	2	2	no	—
183	C₂×Dic₆₇	—	4	1	no	—
184	C₂×Dic₆₇	—	4	1	no	—
185	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₄×D₆₇	—	2	2	no	—
192	C₄×D₆₇	—	2	2	no	—
193	C₄×D₆₇	—	2	2	no	—
194	C₄×D₆₇	—	2	2	no	—
195	C₄×D₆₇	—	2	1	no	—
196	C₄×D₆₇	—	2	1	no	—
197	C₄×D₆₇	—	2	1	no	—
198	C₄×D₆₇	—	2	1	no	—
199	C₄×D₆₇	—	2	2	no	—
200	C₄×D₆₇	—	4	1	no	—
201	C₄×D₆₇	—	4	1	no	—
202	C₄×D₆₇	—	4	1	no	—
203	C₄×D₆₇	—	4	1	no	—
204	C₄×D₆₇	C₂×M₁₆	1	4	yes	—
205	C₈×C₆₇	—	2	2	no	—
206	C₈×C₆₇	—	4	2	no	—
207	C₈×C₆₇	—	4	1	no	—
208	C₈×C₆₇	—	4	1	no	—
209	C₈×C₆₇	C₂×M₁₆	1	4	yes	—
210	C₆₇⋊C₈	—	2	2	no	—
211	C₆₇⋊C₈	—	4	1	no	—
212	C₆₇⋊C₈	—	4	1	no	—
213	C₆₇⋊C₈	—	4	2	no	—
214	C₆₇⋊C₈	C₂×M₁₆	1	4	yes	—
215	C₂²×C₆₇	—	2	1	no	—
216	C₂²×C₆₇	—	2	1	no	—
217	C₂²×C₆₇	—	2	1	no	—
218	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
219	C₂×D₆₇	—	2	1	no	—
220	C₂×D₆₇	—	2	1	no	—
221	C₂×D₆₇	—	2	1	no	—
222	C₂×D₆₇	—	2	1	no	—
223	C₂×D₆₇	—	4	4	no	—
224	C₂×D₆₇	#423	1	2	yes	—
225	C₂×D₆₇	#430	1	2	yes	—
226	C₄×C₆₇	—	2	1	no	—
227	C₄×C₆₇	—	2	1	no	—
228	C₄×C₆₇	—	2	1	no	—
229	C₄×C₆₇	—	2	1	no	—
230	C₄×C₆₇	C₂²×M₁₆	1	2	yes	—
231	C₄×C₆₇	#575	1	1	yes	characteristic
232	C₄×C₆₇	#575	1	1	yes	characteristic
233	Dic₆₇	—	2	1	no	—
234	Dic₆₇	—	2	1	no	—
235	Dic₆₇	—	2	1	no	—
236	Dic₆₇	—	2	1	no	—
237	Dic₆₇	—	2	1	no	—
238	Dic₆₇	#423	1	2	yes	—
239	#17797	—	67	1	no	maximal, Sylow
240	C₂×C₆₇	—	4	1	no	—
241	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
242	C₂×C₆₇	#2911	1	1	yes	characteristic
243	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
244	D₆₇	—	2	2	no	—
245	D₆₇	—	2	2	no	—
246	#2920	—	67	1	no	—
247	C₂×#432	—	67	1	no	—
248	C₂×C₄×M₁₆	—	67	1	no	—
249	#1569	—	67	1	no	—
250	#1569	—	67	1	no	—
251	#1569	—	67	1	no	—
252	#1569	D₆₇	1	1	yes	characteristic, complemented
253	#1568	—	67	2	no	—
254	#1568	—	67	2	no	—
255	#1565	—	67	2	no	—
256	#1565	—	67	2	no	—
257	C₆₇	#17797	1	1	yes	characteristic, complemented, Sylow
258	C₂²×M₁₆	—	67	2	no	—
259	C₂²×C₄²	—	67	1	no	—
260	#432	—	67	1	no	—
261	#432	—	67	1	no	—
262	#432	—	67	1	no	—
263	#432	—	67	2	no	—
264	#432	—	134	1	no	—
265	#432	—	134	1	no	—
266	#432	—	134	1	no	—
267	#432	C₂×D₆₇	1	1	yes	characteristic
268	#431	—	67	2	no	—
269	#431	—	134	1	no	—
270	#423	—	67	1	no	—
271	#423	C₂×D₆₇	1	1	yes	characteristic
272	C₂×C₄⋊C₈	—	67	1	no	—
273	C₂×C₄⋊C₈	—	67	1	no	—
274	#407	—	134	1	no	—
275	C₂×C₂²⋊C₈	—	134	1	no	—
276	C₄×M₁₆	—	67	2	no	—
277	C₄×M₁₆	—	67	1	no	—
278	C₄×M₁₆	—	67	1	no	—
279	C₄×M₁₆	—	67	1	no	—
280	C₄×M₁₆	—	67	2	no	—
281	C₄×M₁₆	—	134	1	no	—
282	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
283	C₂×#97	—	67	1	no	—
284	C₂×#97	—	67	1	no	—
285	C₂×#97	—	134	1	no	—
286	C₂×C₄×C₈	—	67	2	no	—
287	#335	—	67	2	no	—
288	#335	—	67	2	no	—
289	#335	—	67	2	no	—
290	#335	C₂×D₆₇	1	2	yes	complemented
291	#334	—	67	2	no	—
292	#334	—	67	2	no	—
293	#334	—	67	2	no	—
294	#334	C₂×D₆₇	1	2	yes	complemented
295	C₂³×C₄	—	67	1	no	—
296	C₂³×C₄	—	67	1	no	—
297	C₂³×C₄	—	67	1	no	—
298	C₂×M₁₆	—	2	1	no	—
299	C₂×M₁₆	—	67	2	no	—
300	C₂×M₁₆	—	67	2	no	—
301	C₂×M₁₆	—	67	2	no	—
302	C₂×M₁₆	—	134	8	no	—
303	C₂×M₁₆	—	134	1	no	—
304	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
305	C₂²×C₈	—	67	2	no	—
306	C₂²×C₈	—	67	2	no	—
307	C₂²×C₈	—	134	1	no	—
308	C₂²×C₈	—	134	1	no	—
309	C₂×C₄²	—	67	1	no	—
310	C₂×C₄²	—	67	1	no	—
311	C₂×C₄²	—	67	1	no	—
312	C₂×C₄²	—	67	1	no	—
313	C₂×C₄²	—	67	1	no	—
314	C₂×C₄²	—	67	1	no	—
315	C₂×C₄²	—	67	1	no	—
316	C₂×C₄²	—	67	1	no	—
317	C₂×C₄²	—	67	1	no	—
318	C₂×C₄²	—	67	1	no	—
319	C₂×C₄²	—	67	1	no	—
320	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
321	C₄⋊C₈	—	2	1	no	—
322	C₄⋊C₈	—	67	1	no	—
323	C₄⋊C₈	—	67	1	no	—
324	C₄⋊C₈	—	67	2	no	—
325	C₄⋊C₈	—	134	1	no	—
326	C₄⋊C₈	—	134	1	no	—
327	C₄⋊C₈	—	134	1	no	—
328	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
329	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
330	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
331	C₂²⋊C₈	—	2	1	no	—
332	C₂²⋊C₈	—	134	1	no	—
333	C₂²⋊C₈	—	134	1	no	—
334	C₂²⋊C₈	—	134	2	no	—
335	C₂²⋊C₈	—	134	2	no	—
336	C₂²⋊C₈	—	134	1	no	—
337	#97	—	2	1	no	—
338	#97	—	67	2	no	—
339	#97	—	67	2	no	—
340	#97	—	67	1	no	—
341	#97	—	67	1	no	—
342	#97	—	134	1	no	—
343	#97	—	134	1	no	—
344	#97	—	134	2	no	—
345	#97	—	134	1	no	—
346	#97	C₄×D₆₇	1	1	yes	characteristic
347	#97	C₄×D₆₇	1	1	yes	characteristic
348	C₄×C₈	—	67	2	no	—
349	C₄×C₈	—	67	2	no	—
350	C₄×C₈	—	67	2	no	—
351	C₄×C₈	—	134	2	no	—
352	C₄×C₈	C₂²×D₆₇	1	2	yes	—
353	C₂⁴	—	67	1	no	—
354	C₂²×C₄	—	268	1	no	—
355	C₂²×C₄	—	268	1	no	—
356	C₂²×C₄	—	268	2	no	—
357	C₂²×C₄	—	67	1	no	—
358	C₂²×C₄	—	67	1	no	—
359	C₂²×C₄	—	67	1	no	—
360	C₂²×C₄	—	67	1	no	—
361	C₂²×C₄	—	67	1	no	—
362	C₂²×C₄	—	67	1	no	—
363	C₂²×C₄	—	67	1	no	—
364	C₂²×C₄	—	67	1	no	—
365	C₂²×C₄	—	67	1	no	—
366	C₂²×C₄	—	67	1	no	—
367	C₂²×C₄	—	67	1	no	—
368	C₂²×C₄	—	67	1	no	—
369	C₂²×C₄	—	67	1	no	—
370	C₂²×C₄	—	67	1	no	—
371	C₂²×C₄	—	67	1	no	—
372	C₂²×C₄	—	134	4	no	—
373	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
374	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
375	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
376	M₁₆	—	2	4	no	—
377	M₁₆	—	4	2	no	—
378	M₁₆	—	268	2	no	—
379	M₁₆	—	67	8	no	—
380	M₁₆	—	134	4	no	—
381	M₁₆	—	134	4	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	268	2	no	—
386	C₂×C₈	—	268	2	no	—
387	C₂×C₈	—	67	2	no	—
388	C₂×C₈	—	67	2	no	—
389	C₂×C₈	—	134	4	no	—
390	C₂×C₈	—	134	2	no	—
391	C₂×C₈	—	134	1	no	—
392	C₂×C₈	—	134	1	no	—
393	C₂×C₈	—	134	4	no	—
394	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
395	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
396	C₄²	—	67	1	no	—
397	C₄²	—	67	1	no	—
398	C₄²	—	67	1	no	—
399	C₄²	—	67	1	no	—
400	C₄²	—	67	1	no	—
401	C₄²	—	67	1	no	—
402	C₄²	—	67	1	no	—
403	C₄²	—	67	1	no	—
404	C₄²	—	67	1	no	—
405	C₄²	—	67	1	no	—
406	C₄²	—	67	1	no	—
407	C₄²	—	67	1	no	—
408	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
409	C₄²	D₄×D₆₇	1	1	yes	characteristic
410	C₄²	Q₈×D₆₇	1	1	yes	characteristic
411	C₄²	C₂³×D₆₇	1	1	yes	characteristic
412	C₂³	—	268	1	no	—
413	C₂³	—	67	1	no	—
414	C₂³	—	67	1	no	—
415	C₂³	—	134	2	no	—
416	C₂³	—	134	2	no	—
417	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	268	1	no	—
423	C₂×C₄	—	268	1	no	—
424	C₂×C₄	—	268	1	no	—
425	C₂×C₄	—	268	1	no	—
426	C₂×C₄	—	268	1	no	—
427	C₂×C₄	—	268	1	no	—
428	C₂×C₄	—	67	1	no	—
429	C₂×C₄	—	67	4	no	—
430	C₂×C₄	—	67	1	no	—
431	C₂×C₄	—	67	1	no	—
432	C₂×C₄	—	67	1	no	—
433	C₂×C₄	—	67	1	no	—
434	C₂×C₄	—	67	1	no	—
435	C₂×C₄	—	134	2	no	—
436	C₂×C₄	—	134	2	no	—
437	C₂×C₄	—	134	2	no	—
438	C₂×C₄	—	134	2	no	—
439	C₂×C₄	—	134	2	no	—
440	C₂×C₄	—	134	2	no	—
441	C₂×C₄	—	134	1	no	—
442	C₂×C₄	—	134	1	no	—
443	C₂×C₄	—	134	1	no	—
444	C₂×C₄	—	134	1	no	—
445	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
446	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
447	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
448	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
449	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
450	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
451	C₈	—	2	2	no	—
452	C₈	—	4	1	no	—
453	C₈	—	4	2	no	—
454	C₈	—	4	1	no	—
455	C₈	—	268	1	no	—
456	C₈	—	268	1	no	—
457	C₈	—	268	2	no	—
458	C₈	—	67	4	no	—
459	C₈	—	134	2	no	—
460	C₈	M₁₆×D₆₇	1	4	yes	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	2	1	no	—
464	C₂²	—	268	4	no	—
465	C₂²	—	67	2	no	—
466	C₂²	—	67	2	no	—
467	C₂²	—	134	1	no	—
468	C₂²	—	134	1	no	—
469	C₂²	—	134	1	no	—
470	C₂²	—	134	1	no	—
471	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
472	C₄	—	2	1	no	—
473	C₄	—	2	1	no	—
474	C₄	—	2	1	no	—
475	C₄	—	2	1	no	—
476	C₄	—	67	2	no	—
477	C₄	—	134	1	no	—
478	C₄	—	134	1	no	—
479	C₄	—	134	1	no	—
480	C₄	—	134	1	no	—
481	C₄	—	134	1	no	—
482	C₄	C₂×M₁₆×D₆₇	1	2	yes	—
483	C₄	#16864192	1	1	yes	characteristic
484	C₄	#16864195	1	1	yes	characteristic
485	C₂	—	4	1	no	—
486	C₂	—	134	2	no	—
487	C₂	—	134	2	no	—
488	C₂	#423×D₆₇	1	1	yes	characteristic
489	C₂	#22568796	1	1	yes	characteristic
490	C₂	#430×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1569	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₆₇	C₄⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1569	D₆₇
hypocenter	C₆₇	#17797
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1569

Complements

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

N	H ≅ G/N	type
4. #22558977	C₂	semidirect
5. #22558976	C₂	semidirect
6. #22558974	C₂	semidirect
7. #22558971	C₂	semidirect
8. #22558969	C₂	semidirect
9. #22558960	C₂	semidirect
10. #22558958	C₂	semidirect
11. C₆₇×#1569	C₂	semidirect
38. #16863149	C₂²	semidirect
39. #16863148	C₂²	semidirect
40. #16863148	C₂²	semidirect
41. #16863147	C₂²	semidirect
42. #16863146	C₂²	semidirect
43. #16863146	C₂²	semidirect
47. #335×C₆₇	C₂²	semidirect
48. #334×C₆₇	C₂²	semidirect
130. #11761611	M₁₆	semidirect
252. #1569	D₆₇	semidirect
257. C₆₇	#17797	semidirect
290. #335	C₂×D₆₇	semidirect
294. #334	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568841	2
#432×D₆₇	2
C₄×M₁₆×D₆₇	2
#22558977	2
#22558976	2
#22558974	2
#22558971	2
#22558969	2
#22558960	2
#22558958	2
C₆₇×#1569	2
#17797	67