Subgroups — #30914725

Order 17152.

690 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570067	C₂	1	2	yes	maximal
2	#22568801	C₂	1	1	yes	maximal, characteristic
3	#22561722	C₂	1	2	yes	maximal, complemented
4	#22561551	C₂	1	2	yes	maximal, complemented
5	#22560989	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560958	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560432	C₂	1	2	yes	maximal
8	#22560364	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560269	C₂	1	1	yes	maximal, characteristic
10	#22560048	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#3163	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524049	C₂²	1	1	yes	characteristic
13	#16863876	C₂²	1	2	yes	—
14	#16863700	—	2	2	no	—
15	#16863693	—	2	2	no	—
16	#16863683	—	2	2	no	—
17	#16863678	—	2	2	no	—
18	C₂×#14523979	C₂²	1	1	yes	characteristic
19	#16863439	C₂²	1	2	yes	—
20	#16863439	C₂²	1	2	yes	—
21	#16863437	—	2	2	no	—
22	#16863431	—	2	1	no	—
23	C₂×#14523975	C₂²	1	1	yes	characteristic
24	C₂×#14523974	C₂²	1	2	yes	—
25	#106×D₆₇	—	2	1	no	—
26	#16863423	C₂²	1	2	yes	—
27	C₂×#14523969	C₂²	1	1	yes	characteristic
28	#16863395	C₂²	1	2	yes	—
29	#16863390	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
31	#16863265	C₂²	1	2	yes	—
32	#16863165	C₂²	1	2	yes	—
33	#16863164	C₂²	1	2	yes	—
34	#16863160	C₂²	1	1	yes	characteristic
35	#16863155	C₂²	1	1	yes	characteristic
36	#16863155	C₂²	1	2	yes	—
37	#544×C₆₇	C₂²	1	2	yes	—
38	#501×C₆₇	—	2	2	no	—
39	#499×C₆₇	—	2	1	no	—
40	#498×C₆₇	—	2	1	no	—
41	#428×C₆₇	C₂²	1	2	yes	—
42	#427×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
44	#405×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#11761628	C₂³	1	1	yes	characteristic
46	#14524049	—	2	2	no	—
47	#14524049	Q₈	1	4	yes	complemented
48	C₂×#11761612	C₂³	1	2	yes	—
49	C₂×#11761612	C₂³	1	1	yes	characteristic
50	C₂×#11761611	—	2	1	no	—
51	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×C₈×D₆₇	—	2	1	no	—
53	C₄×Dic₁₃₄	—	2	2	no	—
54	C₄×D₂₆₈	—	2	2	no	—
55	(C₂²⋊D₆₇)⋊C₄	—	4	2	no	—
56	#14524001	—	2	2	no	—
57	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
58	C₂×Dic₆₇⋊C₄	—	4	2	no	—
59	C₄⋊C₄×D₆₇	—	2	2	no	—
60	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
61	#14523979	—	2	2	no	—
62	#14523976	—	2	2	no	—
63	#14523976	—	2	2	no	—
64	#14523975	—	2	1	no	—
65	#14523975	—	2	1	no	—
66	#14523974	—	2	2	no	—
67	#14523974	—	2	2	no	—
68	#14523974	—	2	2	no	—
69	#14523974	—	2	2	no	—
70	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
71	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
72	#14523969	Q₈	1	4	yes	complemented
73	#14523966	—	2	2	no	—
74	#14523966	D₄	1	2	yes	—
75	#14523963	—	2	2	no	—
76	#14523963	D₄	1	2	yes	—
77	C₈×Dic₆₇	—	2	2	no	—
78	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄⋊Q₈×C₆₇	—	2	2	no	—
81	C₄⋊Q₈×C₆₇	—	2	2	no	—
82	#125×C₆₇	—	2	2	no	—
83	#125×C₆₇	—	2	2	no	—
84	C₂²⋊Q₈×C₆₇	—	4	2	no	—
85	#117×C₆₇	C₂³	1	1	yes	characteristic
86	#117×C₆₇	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
88	#107×C₆₇	—	2	2	no	—
89	#107×C₆₇	D₄	1	2	yes	—
90	#106×C₆₇	—	2	1	no	—
91	#106×C₆₇	—	2	1	no	—
92	#106×C₆₇	—	2	2	no	—
93	#106×C₆₇	D₄	1	1	yes	characteristic
94	#106×C₆₇	D₄	1	1	yes	characteristic
95	#97×C₆₇	—	2	1	no	—
96	C₄×C₈×C₆₇	—	2	1	no	—
97	#11761628	—	2	1	no	—
98	#11761628	—	2	1	no	—
99	#11761628	C₂×Q₈	1	2	yes	—
100	#11761628	C₂×Q₈	1	2	yes	—
101	C₂×Dic₁₃₄	C₂×D₄	1	1	yes	characteristic
102	C₂×D₂₆₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₂²⋊D₆₇	—	2	1	no	—
104	C₂²×Dic₆₇	—	2	1	no	—
105	C₂²×Dic₆₇	—	2	2	no	—
106	C₂×C₄×D₆₇	—	2	1	no	—
107	C₂×C₄×D₆₇	—	4	2	no	—
108	#11761612	—	2	1	no	—
109	#11761612	—	2	2	no	—
110	#11761612	—	2	2	no	—
111	#11761612	C₂×Q₈	1	2	yes	—
112	#11761611	—	2	4	no	—
113	C₂×C₆₇⋊C₈	—	2	1	no	—
114	C₂×C₆₇⋊C₈	—	2	1	no	—
115	C₂×C₆₇⋊C₈	—	2	1	no	—
116	C₂×C₆₇⋊C₈	—	2	2	no	—
117	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
118	C₈×D₆₇	—	2	4	no	—
119	C₄⋊Dic₆₇	—	2	2	no	—
120	C₄⋊Dic₆₇	—	2	2	no	—
121	Dic₆₇⋊C₄	—	4	2	no	—
122	Dic₆₇⋊C₄	—	8	4	no	—
123	(C₂×D₆₇)⋊C₄	—	4	2	no	—
124	C₄×Dic₆₇	—	2	1	no	—
125	C₄×Dic₆₇	—	4	2	no	—
126	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
127	C₂×Q₈×C₆₇	—	4	2	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
130	M₁₆×C₆₇	—	2	1	no	—
131	M₁₆×C₆₇	C₂×Q₈	1	2	yes	—
132	C₂×C₈×C₆₇	—	2	1	no	—
133	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
136	C₂×C₈×C₆₇	#41	1	2	yes	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	4	2	no	—
140	C₄⋊C₄×C₆₇	—	4	2	no	—
141	C₄⋊C₄×C₆₇	—	4	2	no	—
142	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
143	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
144	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
145	C₂²⋊C₄×C₆₇	—	2	1	no	—
146	C₂²⋊C₄×C₆₇	—	2	2	no	—
147	C₄²×C₆₇	—	2	1	no	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂²×D₆₇	—	2	1	no	—
151	Q₈×C₆₇	—	16	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	—	8	2	no	—
157	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
163	C₂²⋊D₆₇	—	2	2	no	—
164	C₂²⋊D₆₇	—	2	2	no	—
165	C₂×Dic₆₇	—	2	4	no	—
166	C₂×Dic₆₇	—	2	1	no	—
167	C₂×Dic₆₇	—	2	2	no	—
168	C₂×Dic₆₇	—	4	2	no	—
169	C₂×Dic₆₇	—	8	2	no	—
170	D₂₆₈	—	2	1	no	—
171	D₂₆₈	C₄⋊Q₈	1	2	yes	—
172	C₄×D₆₇	—	2	2	no	—
173	C₄×D₆₇	—	2	2	no	—
174	C₄×D₆₇	—	16	2	no	—
175	Dic₁₃₄	—	2	1	no	—
176	Dic₁₃₄	C₄⋊Q₈	1	2	yes	—
177	C₈×C₆₇	—	2	2	no	—
178	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
179	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
180	C₆₇⋊C₈	—	2	2	no	—
181	C₆₇⋊C₈	—	2	2	no	—
182	C₆₇⋊C₈	—	4	2	no	—
183	C₂²×C₆₇	—	2	2	no	—
184	C₂²×C₆₇	#544	1	1	yes	characteristic
185	C₂²×C₆₇	D₄²	1	1	yes	characteristic
186	C₂²×C₆₇	#554	1	1	yes	characteristic
187	C₂×D₆₇	—	2	2	no	—
188	C₂×D₆₇	—	4	2	no	—
189	C₄×C₆₇	—	4	2	no	—
190	C₄×C₆₇	—	4	2	no	—
191	C₄×C₆₇	—	4	2	no	—
192	C₄×C₆₇	—	8	2	no	—
193	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
194	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
195	C₄×C₆₇	#547	1	1	yes	characteristic
196	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
197	Dic₆₇	—	2	4	no	—
198	Dic₆₇	—	2	2	no	—
199	Dic₆₇	—	8	2	no	—
200	#26081	—	67	1	no	maximal, Sylow
201	C₂×C₆₇	—	2	1	no	—
202	C₂×C₆₇	—	2	1	no	—
203	C₂×C₆₇	#2440	1	1	yes	characteristic
204	C₂×C₆₇	#3291	1	2	yes	—
205	D₆₇	—	4	2	no	—
206	#3163	D₆₇	1	1	yes	characteristic, complemented
207	#3084	—	67	2	no	—
208	#2912	—	67	1	no	—
209	#2090	—	67	2	no	—
210	#2060	—	67	2	no	—
211	#1951	—	67	1	no	—
212	#1945	—	67	1	no	—
213	#1859	—	67	2	no	—
214	#1844	—	67	1	no	—
215	#1815	—	67	1	no	—
216	#1764	—	67	1	no	—
217	C₆₇	#26081	1	1	yes	characteristic, complemented, Sylow
218	C₂×#131	—	67	1	no	—
219	#544	C₂×D₆₇	1	2	yes	—
220	#520	—	67	2	no	—
221	#501	—	2	2	no	—
222	#499	—	2	1	no	—
223	#498	—	2	1	no	—
224	#479	—	134	2	no	—
225	#478	—	134	2	no	—
226	#476	—	134	2	no	—
227	#475	—	134	2	no	—
228	C₂×#108	—	67	1	no	—
229	#428	—	67	2	no	—
230	#428	—	67	2	no	—
231	#428	—	134	2	no	—
232	#428	C₂×D₆₇	1	2	yes	—
233	#427	—	134	1	no	—
234	#427	C₂×D₆₇	1	1	yes	characteristic
235	C₂×#106	—	67	1	no	—
236	C₂×#106	—	67	2	no	—
237	C₂×#106	—	134	1	no	—
238	C₂×#106	C₂×D₆₇	1	1	yes	characteristic
239	#424	—	67	2	no	—
240	C₂×#104	—	67	1	no	—
241	#417	—	67	2	no	—
242	#416	—	67	1	no	—
243	#405	C₂×D₆₇	1	1	yes	characteristic
244	C₂×C₄×C₈	—	67	1	no	—
245	#384	—	67	2	no	—
246	#343	—	67	2	no	—
247	#343	—	67	2	no	—
248	#341	—	67	1	no	—
249	#339	—	67	1	no	—
250	#339	—	67	2	no	—
251	C₂×#41	—	67	1	no	—
252	#131	—	67	4	no	—
253	#131	—	134	2	no	—
254	C₂×M₁₆	—	67	1	no	—
255	C₂×M₁₆	—	67	2	no	—
256	C₂×M₁₆	—	134	1	no	—
257	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
258	C₂²×C₈	—	67	1	no	—
259	C₂²×C₈	—	134	1	no	—
260	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
261	C₄⋊Q₈	—	2	2	no	—
262	C₄⋊Q₈	—	2	2	no	—
263	#125	—	2	2	no	—
264	#125	—	2	2	no	—
265	C₂²⋊Q₈	—	4	2	no	—
266	C₄×Q₈	—	134	2	no	—
267	C₄×D₄	—	268	2	no	—
268	C₄×D₄	—	134	2	no	—
269	#117	—	134	2	no	—
270	#117	C₂²×D₆₇	1	1	yes	characteristic
271	#117	C₂²×D₆₇	1	2	yes	—
272	C₂×C₄⋊C₄	—	268	2	no	—
273	C₂×C₄⋊C₄	—	67	2	no	—
274	C₂×C₄⋊C₄	—	134	2	no	—
275	C₂×C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
276	C₂×C₄²	—	67	1	no	—
277	#108	—	134	2	no	—
278	#107	—	2	2	no	—
279	#107	—	134	2	no	—
280	#107	—	134	2	no	—
281	#107	C₂²⋊D₆₇	1	2	yes	—
282	#106	—	2	1	no	—
283	#106	—	2	1	no	—
284	#106	—	2	2	no	—
285	#106	—	134	2	no	—
286	#106	—	134	2	no	—
287	#106	—	134	1	no	—
288	#106	—	134	1	no	—
289	#106	—	134	2	no	—
290	#106	—	134	2	no	—
291	#106	C₂²⋊D₆₇	1	1	yes	characteristic
292	#106	C₂²⋊D₆₇	1	1	yes	characteristic
293	C₄⋊C₈	—	134	2	no	—
294	C₄⋊C₈	—	134	2	no	—
295	#104	—	67	4	no	—
296	#103	—	67	2	no	—
297	#103	—	134	2	no	—
298	#102	—	67	2	no	—
299	#102	—	134	2	no	—
300	#97	—	2	1	no	—
301	C₄×C₈	—	2	1	no	—
302	C₄×C₈	—	134	2	no	—
303	#41	—	67	2	no	—
304	#41	—	67	2	no	—
305	#41	—	134	1	no	—
306	#41	—	134	1	no	—
307	C₂×Q₈	—	4	2	no	—
308	C₂×Q₈	—	67	1	no	—
309	C₂×D₄	—	67	1	no	—
310	C₂×D₄	—	134	1	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	268	2	no	—
313	C₂²×C₄	—	134	2	no	—
314	C₂²×C₄	—	134	1	no	—
315	C₂²×C₄	—	134	1	no	—
316	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
317	M₁₆	—	2	1	no	—
318	M₁₆	—	67	2	no	—
319	M₁₆	—	134	2	no	—
320	M₁₆	—	134	2	no	—
321	M₁₆	—	134	1	no	—
322	M₁₆	—	134	4	no	—
323	M₁₆	Q₈×D₆₇	1	2	yes	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	67	2	no	—
326	C₂×C₈	—	134	2	no	—
327	C₂×C₈	—	134	1	no	—
328	C₂×C₈	—	134	1	no	—
329	C₂×C₈	—	134	1	no	—
330	C₂×C₈	—	134	4	no	—
331	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
332	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
333	C₂×C₈	Q₈×D₆₇	1	2	yes	—
334	C₂×C₈	#11761629	1	2	yes	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	4	2	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	—	4	2	no	—
340	C₄⋊C₄	—	268	2	no	—
341	C₄⋊C₄	—	536	4	no	—
342	C₄⋊C₄	—	134	2	no	—
343	C₄⋊C₄	—	134	2	no	—
344	C₄⋊C₄	D₄×D₆₇	1	2	yes	—
345	C₄⋊C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
346	C₄⋊C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
347	C₂²⋊C₄	—	2	2	no	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	268	2	no	—
350	C₄²	—	2	2	no	—
351	C₄²	—	2	1	no	—
352	C₄²	—	268	2	no	—
353	C₄²	—	67	2	no	—
354	C₄²	—	134	1	no	—
355	C₂³	—	134	1	no	—
356	C₂³	#14524088	1	1	yes	characteristic
357	Q₈	—	16	2	no	—
358	Q₈	—	67	2	no	—
359	Q₈	—	134	1	no	—
360	D₄	—	67	2	no	—
361	D₄	—	134	2	no	—
362	D₄	—	134	2	no	—
363	D₄	—	134	1	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	—	268	2	no	—
370	C₂×C₄	—	536	2	no	—
371	C₂×C₄	—	1072	2	no	—
372	C₂×C₄	—	134	4	no	—
373	C₂×C₄	—	134	1	no	—
374	C₂×C₄	—	134	2	no	—
375	C₂×C₄	—	134	2	no	—
376	C₂×C₄	—	134	2	no	—
377	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
378	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
379	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
380	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
381	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
382	C₂×C₄	#14524091	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	268	2	no	—
385	C₈	—	134	2	no	—
386	C₈	—	134	2	no	—
387	C₈	#14524019	1	2	yes	—
388	C₈	#14524019	1	2	yes	—
389	C₂²	—	2	2	no	—
390	C₂²	—	268	2	no	—
391	C₂²	—	134	2	no	—
392	C₂²	#16864023	1	1	yes	characteristic
393	C₂²	D₄×C₂²⋊D₆₇	1	1	yes	characteristic
394	C₂²	#16864107	1	1	yes	characteristic
395	C₄	—	4	2	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	2	no	—
398	C₄	—	8	2	no	—
399	C₄	—	536	2	no	—
400	C₄	—	134	4	no	—
401	C₄	—	134	2	no	—
402	C₄	C₂×#14524019	1	1	yes	characteristic
403	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
404	C₄	#16864049	1	1	yes	characteristic
405	C₄	Q₈×C₂²⋊D₆₇	1	1	yes	characteristic
406	C₂	—	2	1	no	—
407	C₂	—	2	1	no	—
408	C₂	—	268	2	no	—
409	C₂	#22563907	1	1	yes	characteristic
410	C₂	#22572000	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₆₇
commutator, cosocle	C₂²×C₄×C₆₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#3163	C₂
socle	C₂²×C₆₇	D₄²
prime radical	C₂×#11761628	C₂³
hypercenter	#3163	D₆₇
hypocenter	C₆₇	#26081
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #3163

Complements

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

N	H ≅ G/N	type
3. #22561722	C₂	semidirect
4. #22561551	C₂	semidirect
5. #22560989	C₂	semidirect
6. #22560958	C₂	semidirect
8. #22560364	C₂	semidirect
11. C₆₇×#3163	C₂	semidirect
47. #14524049	Q₈	semidirect
72. #14523969	Q₈	semidirect
206. #3163	D₆₇	semidirect
217. C₆₇	#26081	semidirect

Maximal subgroups

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

subgroup	index
#22570067	2
#22568801	2
#22561722	2
#22561551	2
#22560989	2
#22560958	2
#22560432	2
#22560364	2
#22560269	2
#22560048	2
C₆₇×#3163	2
#26081	67