Subgroups — #30847144

Order 17152.

690 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22572558	C₂	1	1	yes	maximal, characteristic
2	#22571855	C₂	1	2	yes	maximal, complemented
3	#22559826	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559696	C₂	1	2	yes	maximal, complemented
5	#22559603	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559595	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559458	C₂	1	2	yes	maximal, complemented
8	#22559443	C₂	1	2	yes	maximal, complemented
9	#22559094	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11761615	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524074	—	2	2	no	—
13	#16864051	C₂²	1	2	yes	—
14	#16863801	C₂²	1	1	yes	characteristic
15	#16863799	C₂²	1	2	yes	—
16	#16863775	C₂²	1	2	yes	—
17	#16863708	—	2	2	no	—
18	#16863676	C₂²	1	2	yes	—
19	#16863652	—	2	2	no	—
20	#16863629	—	2	1	no	—
21	#16863606	C₂²	1	2	yes	—
22	#16863598	C₂²	1	2	yes	—
23	#16863556	—	2	2	no	—
24	#16863552	—	2	2	no	—
25	C₄×#11761615	C₂²	1	1	yes	characteristic
26	#16863474	C₂²	1	2	yes	—
27	C₈×D₂₆₈	C₂²	1	1	yes	characteristic
28	#107×D₆₇	—	2	1	no	—
29	#16863149	C₂²	1	1	yes	characteristic
30	#16863146	C₂²	1	1	yes	characteristic
31	#16863142	C₂²	1	2	yes	—
32	#16863141	C₂²	1	2	yes	—
33	Q₈⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
34	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
35	#16863123	C₂²	1	1	yes	characteristic
36	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
37	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
38	#500×C₆₇	—	2	1	no	—
39	#500×C₆₇	C₂²	1	1	yes	characteristic
40	#474×C₆₇	—	2	2	no	—
41	#474×C₆₇	C₂²	1	2	yes	—
42	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
43	C₄×D₈×C₆₇	—	2	1	no	—
44	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#11761628	—	2	2	no	—
46	#14524074	—	4	8	no	—
47	C₂×#11761618	—	2	2	no	—
48	C₂×#11761618	—	2	2	no	—
49	C₂×#11761618	—	2	2	no	—
50	C₂×#11761615	—	2	2	no	—
51	C₂×#11761615	D₄	1	1	yes	characteristic
52	C₂×#11761612	—	2	2	no	—
53	C₂×C₈×D₆₇	—	2	1	no	—
54	#14524043	C₂³	1	2	yes	—
55	#14524025	—	4	2	no	—
56	#14524022	—	2	2	no	—
57	#14524014	—	2	2	no	—
58	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×D₆₇	—	2	2	no	—
60	#14523977	—	2	1	no	—
61	#14523977	—	2	1	no	—
62	#14523976	—	2	2	no	—
63	#14523976	—	4	2	no	—
64	#14523974	D₄	1	1	yes	characteristic
65	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
66	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
67	#14523967	—	2	2	no	—
68	#14523967	D₄	1	1	yes	characteristic
69	#14523966	—	2	2	no	—
70	#14523964	—	2	2	no	—
71	#14523963	D₄	1	1	yes	characteristic
72	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
73	(C₂×D₆₇)⋊C₈	—	2	1	no	—
74	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₆₇	—	2	2	no	—
77	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
78	C₄⋊Q₈×C₆₇	—	2	2	no	—
79	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
80	C₂²⋊Q₈×C₆₇	—	2	2	no	—
81	C₂²⋊Q₈×C₆₇	—	4	2	no	—
82	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
83	C₄×D₄×C₆₇	—	2	2	no	—
84	C₄×D₄×C₆₇	C₂³	1	2	yes	—
85	#107×C₆₇	—	2	1	no	—
86	#107×C₆₇	—	2	1	no	—
87	#107×C₆₇	—	2	2	no	—
88	#107×C₆₇	D₄	1	1	yes	characteristic
89	C₄⋊C₈×C₆₇	—	2	1	no	—
90	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
91	#102×C₆₇	—	2	2	no	—
92	#102×C₆₇	D₄	1	2	yes	—
93	C₄×C₈×C₆₇	—	2	1	no	—
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	#11761628	—	4	4	no	—
96	#11761628	—	8	2	no	—
97	C₂×Dic₁₃₄	—	2	2	no	—
98	C₂×D₂₆₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₂²⋊D₆₇	—	4	2	no	—
100	C₂×C₄×D₆₇	—	2	1	no	—
101	C₂×C₄×D₆₇	—	4	2	no	—
102	#11761618	—	4	4	no	—
103	#11761618	—	4	4	no	—
104	#11761618	—	8	2	no	—
105	#11761615	—	4	4	no	—
106	#11761615	D₈	1	4	yes	complemented
107	#11761612	—	4	4	no	—
108	C₂×C₆₇⋊C₈	—	2	1	no	—
109	C₂×C₆₇⋊C₈	—	2	1	no	—
110	C₂×C₆₇⋊C₈	—	2	2	no	—
111	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
113	C₈×D₆₇	—	4	2	no	—
114	C₄⋊Dic₆₇	—	2	2	no	—
115	C₄⋊Dic₆₇	—	2	2	no	—
116	C₄⋊Dic₆₇	C₂×D₄	1	1	yes	characteristic
117	Dic₆₇⋊C₄	—	4	2	no	—
118	C₂²⋊Dic₆₇	—	4	2	no	—
119	(C₂×D₆₇)⋊C₄	—	2	1	no	—
120	C₂×Q₈×C₆₇	—	2	2	no	—
121	C₂×Q₈×C₆₇	—	4	2	no	—
122	C₂×Q₈×C₆₇	—	4	4	no	—
123	C₂×Q₈×C₆₇	—	8	2	no	—
124	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	4	2	no	—
128	D₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₆₇	—	2	1	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	4	2	no	—
136	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
138	C₂²⋊C₄×C₆₇	—	2	2	no	—
139	C₂²⋊C₄×C₆₇	—	4	2	no	—
140	C₄²×C₆₇	—	2	1	no	—
141	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₂³×C₆₇	—	2	2	no	—
143	C₂²×D₆₇	—	2	1	no	—
144	Q₈×C₆₇	—	2	1	no	—
145	Q₈×C₆₇	—	4	2	no	—
146	Q₈×C₆₇	—	4	2	no	—
147	Q₈×C₆₇	—	8	4	no	—
148	Q₈×C₆₇	C₂×D₈	1	2	yes	—
149	D₄×C₆₇	—	2	2	no	—
150	D₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	1	no	—
152	C₂×C₄×C₆₇	—	2	1	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	8	4	no	—
158	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²⋊D₆₇	—	8	4	no	—
162	C₂×Dic₆₇	—	2	1	no	—
163	C₂×Dic₆₇	—	4	2	no	—
164	D₂₆₈	—	2	1	no	—
165	D₂₆₈	C₂×D₈	1	2	yes	—
166	C₄×D₆₇	—	4	1	no	—
167	C₄×D₆₇	—	4	1	no	—
168	C₄×D₆₇	—	8	4	no	—
169	Dic₁₃₄	—	4	2	no	—
170	Dic₁₃₄	—	4	2	no	—
171	C₈×C₆₇	—	2	1	no	—
172	C₈×C₆₇	—	4	1	no	—
173	C₈×C₆₇	#137	1	2	yes	complemented
174	C₆₇⋊C₈	—	2	2	no	—
175	C₆₇⋊C₈	—	2	1	no	—
176	C₆₇⋊C₈	—	4	2	no	—
177	C₆₇⋊C₈	—	4	1	no	—
178	C₆₇⋊C₈	C₂×D₈	1	2	yes	—
179	C₂²×C₆₇	—	4	2	no	—
180	C₂²×C₆₇	—	4	2	no	—
181	C₂²×C₆₇	—	4	2	no	—
182	C₂²×C₆₇	D₄²	1	1	yes	characteristic
183	C₂×D₆₇	—	2	2	no	—
184	C₂×D₆₇	—	4	1	no	—
185	C₂×D₆₇	—	4	1	no	—
186	C₄×C₆₇	—	2	2	no	—
187	C₄×C₆₇	—	2	1	no	—
188	C₄×C₆₇	—	4	2	no	—
189	C₄×C₆₇	—	4	1	no	—
190	C₄×C₆₇	—	8	2	no	—
191	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
192	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
193	C₄×C₆₇	#576	1	1	yes	characteristic
194	C₄×C₆₇	#578	1	1	yes	characteristic
195	Dic₆₇	—	4	1	no	—
196	Dic₆₇	—	8	2	no	—
197	#19887	—	67	1	no	maximal, Sylow
198	C₂×C₆₇	—	8	2	no	—
199	C₂×C₆₇	#3276	1	1	yes	characteristic
200	C₂×C₆₇	D₄×D₈	1	1	yes	characteristic
201	C₂×C₆₇	#3293	1	1	yes	characteristic
202	D₆₇	—	4	2	no	—
203	Q₈×D₈	D₆₇	1	1	yes	characteristic, complemented
204	#3334	—	67	1	no	—
205	#3276	—	67	2	no	—
206	#1713	—	67	1	no	—
207	#1689	—	67	2	no	—
208	#1674	—	67	1	no	—
209	#1673	—	67	1	no	—
210	#1649	—	67	2	no	—
211	#1647	—	67	2	no	—
212	#1589	—	67	1	no	—
213	C₈×SD₁₆	—	67	1	no	—
214	C₆₇	#19887	1	1	yes	characteristic, complemented, Sylow
215	C₂×#137	—	134	2	no	—
216	D₄×Q₈	C₂×D₆₇	1	2	yes	—
217	#547	—	67	2	no	—
218	#500	—	2	1	no	—
219	#500	—	67	2	no	—
220	#500	—	67	1	no	—
221	#500	C₂×D₆₇	1	1	yes	characteristic
222	#494	—	67	2	no	—
223	#480	—	134	2	no	—
224	#474	—	2	2	no	—
225	#474	—	67	2	no	—
226	#474	C₂×D₆₇	1	2	yes	—
227	#470	—	134	2	no	—
228	#466	—	134	1	no	—
229	#462	—	67	2	no	—
230	#461	—	67	2	no	—
231	#452	—	134	2	no	—
232	#451	—	134	2	no	—
233	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
234	C₄×SD₁₆	—	67	1	no	—
235	C₄×D₈	—	2	1	no	—
236	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
237	#436	—	67	2	no	—
238	C₈×D₄	—	67	1	no	—
239	C₂×#107	—	134	1	no	—
240	#335	—	67	1	no	—
241	#334	—	67	1	no	—
242	#332	—	67	2	no	—
243	#332	—	67	2	no	—
244	Q₈⋊C₈	—	67	1	no	—
245	D₄⋊C₈	—	67	2	no	—
246	D₄⋊C₈	—	67	1	no	—
247	C₈²	—	67	1	no	—
248	C₂×#41	—	134	2	no	—
249	C₂²×Q₈	—	2	2	no	—
250	#137	—	268	8	no	—
251	C₂×Q₁₆	—	134	2	no	—
252	C₂×Q₁₆	—	134	2	no	—
253	C₂×Q₁₆	—	134	2	no	—
254	C₂×SD₁₆	—	67	1	no	—
255	C₂×SD₁₆	—	134	2	no	—
256	C₂×D₈	C₂²⋊D₆₇	1	1	yes	characteristic
257	C₂×M₁₆	—	134	2	no	—
258	C₂²×C₈	—	134	1	no	—
259	C₄⋊Q₈	—	2	2	no	—
260	C₄⋊Q₈	—	67	2	no	—
261	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
262	#123	—	268	2	no	—
263	C₂²⋊Q₈	—	2	2	no	—
264	C₂²⋊Q₈	—	4	2	no	—
265	C₂²⋊Q₈	—	134	2	no	—
266	#121	—	134	2	no	—
267	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
268	C₄×D₄	—	2	2	no	—
269	C₄×D₄	—	67	1	no	—
270	C₄×D₄	C₂²×D₆₇	1	2	yes	—
271	C₂×C₄⋊C₄	—	134	2	no	—
272	#107	—	2	1	no	—
273	#107	—	2	1	no	—
274	#107	—	2	2	no	—
275	#107	—	268	2	no	—
276	#107	—	134	1	no	—
277	#107	—	134	1	no	—
278	#107	—	134	2	no	—
279	#107	C₂²⋊D₆₇	1	1	yes	characteristic
280	#106	—	67	1	no	—
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	67	2	no	—
283	C₄⋊C₈	—	67	1	no	—
284	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
285	#103	—	67	1	no	—
286	#103	—	134	2	no	—
287	#103	—	134	2	no	—
288	#102	—	2	2	no	—
289	#102	—	67	1	no	—
290	#102	—	134	2	no	—
291	#102	C₂²⋊D₆₇	1	2	yes	—
292	C₂²⋊C₈	—	134	2	no	—
293	C₂²⋊C₈	—	134	1	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	—	67	1	no	—
296	C₄×C₈	—	67	1	no	—
297	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
298	#41	—	268	4	no	—
299	#41	—	536	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	4	2	no	—
302	C₂×Q₈	—	4	4	no	—
303	C₂×Q₈	—	8	2	no	—
304	C₂×Q₈	—	134	2	no	—
305	C₂×Q₈	D₄×D₆₇	1	1	yes	characteristic
306	C₂×D₄	—	268	2	no	—
307	C₂×D₄	—	67	1	no	—
308	C₂×D₄	C₂×C₂²⋊D₆₇	1	2	yes	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	4	2	no	—
311	C₂²×C₄	—	268	2	no	—
312	C₂²×C₄	—	134	1	no	—
313	Q₁₆	—	268	4	no	—
314	Q₁₆	—	268	4	no	—
315	Q₁₆	—	536	2	no	—
316	SD₁₆	—	268	4	no	—
317	SD₁₆	—	67	4	no	—
318	D₈	—	2	2	no	—
319	M₁₆	—	268	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	268	2	no	—
322	C₂×C₈	—	67	1	no	—
323	C₂×C₈	—	67	1	no	—
324	C₂×C₈	—	134	2	no	—
325	C₂×C₈	—	134	1	no	—
326	C₂×C₈	—	134	1	no	—
327	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
328	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	—	268	2	no	—
334	C₄⋊C₄	—	67	1	no	—
335	C₄⋊C₄	—	134	2	no	—
336	C₄⋊C₄	—	134	2	no	—
337	C₄⋊C₄	D₄×D₆₇	1	1	yes	characteristic
338	C₄⋊C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
339	C₂²⋊C₄	—	2	2	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	268	2	no	—
342	C₂²⋊C₄	—	134	1	no	—
343	C₄²	—	2	1	no	—
344	C₄²	C₂³×D₆₇	1	1	yes	characteristic
345	C₂³	—	2	2	no	—
346	C₂³	—	134	1	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	8	4	no	—
351	Q₈	—	268	2	no	—
352	Q₈	—	268	2	no	—
353	Q₈	D₈×D₆₇	1	2	yes	—
354	D₄	—	2	2	no	—
355	D₄	—	2	2	no	—
356	D₄	—	536	4	no	—
357	D₄	—	67	2	no	—
358	D₄	—	134	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	8	4	no	—
366	C₂×C₄	—	268	1	no	—
367	C₂×C₄	—	268	1	no	—
368	C₂×C₄	—	268	2	no	—
369	C₂×C₄	—	536	4	no	—
370	C₂×C₄	—	134	1	no	—
371	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
372	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
373	C₂×C₄	#14524088	1	1	yes	characteristic
374	C₈	—	2	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	268	2	no	—
377	C₈	—	268	1	no	—
378	C₈	—	67	2	no	—
379	C₈	—	134	2	no	—
380	C₈	—	134	1	no	—
381	C₈	#14524074	1	2	yes	complemented
382	C₂²	—	4	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	268	1	no	—
386	C₂²	—	268	1	no	—
387	C₂²	—	134	2	no	—
388	C₂²	D₄×C₂²⋊D₆₇	1	1	yes	characteristic
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	4	2	no	—
392	C₄	—	4	1	no	—
393	C₄	—	8	2	no	—
394	C₄	—	268	1	no	—
395	C₄	—	536	2	no	—
396	C₄	C₂×D₈×D₆₇	1	1	yes	characteristic
397	C₄	C₂×#14524074	1	1	yes	characteristic
398	C₄	#16864197	1	1	yes	characteristic
399	C₄	#16864211	1	1	yes	characteristic
400	C₂	—	8	2	no	—
401	C₂	—	268	2	no	—
402	C₂	#22571855	1	1	yes	characteristic
403	C₂	D₈×C₂²⋊D₆₇	1	1	yes	characteristic
404	C₂	#22572011	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₆₇
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	Q₈×D₈×C₆₇	C₂
socle	C₂²×C₆₇	D₄²
prime radical	#22572558	C₂
hypercenter	Q₈×D₈	D₆₇
hypocenter	C₆₇	#19887
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈×D₈

Complements

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

N	H ≅ G/N	type
2. #22571855	C₂	semidirect
3. #22559826	C₂	semidirect
4. #22559696	C₂	semidirect
5. #22559603	C₂	semidirect
6. #22559595	C₂	semidirect
7. #22559458	C₂	semidirect
8. #22559443	C₂	semidirect
9. #22559094	C₂	semidirect
10. C₈×#11761615	C₂	semidirect
11. Q₈×D₈×C₆₇	C₂	semidirect
106. #11761615	D₈	semidirect
173. C₈×C₆₇	#137	semidirect
203. Q₈×D₈	D₆₇	semidirect
214. C₆₇	#19887	semidirect
381. C₈	#14524074	semidirect

Maximal subgroups

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

subgroup	index
#22572558	2
#22571855	2
#22559826	2
#22559696	2
#22559603	2
#22559595	2
#22559458	2
#22559443	2
#22559094	2
C₈×#11761615	2
Q₈×D₈×C₆₇	2
#19887	67