Subgroups — #30839684

Order 17152.

602 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570795	C₂	1	1	yes	maximal, characteristic
2	#22570016	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
4	#22559668	C₂	1	4	yes	maximal, complemented
5	#22559614	C₂	1	2	yes	maximal, complemented
6	#22559580	C₂	1	2	yes	maximal, complemented
7	#22558900	C₂	1	2	yes	maximal
8	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic
9	#16863953	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
11	#16863801	C₂²	1	1	yes	characteristic
12	#16863776	C₂²	1	1	yes	characteristic
13	#16863747	C₂²	1	2	yes	—
14	#16863704	—	2	4	no	—
15	#16863692	C₂²	1	4	yes	—
16	#16863682	C₂²	1	4	yes	—
17	#16863637	—	2	1	no	—
18	#16863637	—	2	1	no	—
19	C₈×Dic₁₃₄	—	2	1	no	—
20	C₈×Dic₁₃₄	—	2	1	no	—
21	C₈×Dic₁₃₄	C₂²	1	2	yes	—
22	C₈×Dic₁₃₄	C₂²	1	2	yes	—
23	C₄×Dic₂₆₈	—	2	1	no	—
24	C₄×Dic₂₆₈	—	2	1	no	—
25	#16863438	—	2	2	no	—
26	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
27	C₂×Dic₆₇⋊C₈	—	2	1	no	—
28	#16863406	—	2	2	no	—
29	C₂×C₈×Dic₆₇	—	2	1	no	—
30	#16863144	C₂²	1	4	yes	—
31	#16863141	C₂²	1	4	yes	—
32	#423×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	#335×C₆₇	C₂²	1	2	yes	—
35	#335×C₆₇	C₂²	1	2	yes	—
36	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
37	C₂×Dic₂₆₈	—	2	1	no	—
38	C₂×Dic₂₆₈	—	2	1	no	—
39	C₂²×C₆₇⋊C₈	—	2	1	no	—
40	#14524043	C₂³	1	2	yes	—
41	#14524043	C₂³	1	2	yes	—
42	#14524035	—	2	2	no	—
43	#14524022	—	2	2	no	—
44	#14524022	—	2	2	no	—
45	C₄×Dic₁₃₄	—	2	1	no	—
46	C₄×Dic₁₃₄	—	2	2	no	—
47	C₄×Dic₁₃₄	—	2	1	no	—
48	C₄×Dic₁₃₄	—	2	2	no	—
49	C₄×Dic₁₃₄	D₄	1	2	yes	—
50	C₄×Dic₁₃₄	C₂³	1	2	yes	—
51	#14524002	—	2	2	no	—
52	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
53	C₂×Dic₆₇⋊C₄	—	2	1	no	—
54	C₂×C₄×Dic₆₇	—	2	1	no	—
55	#14523977	—	2	4	no	—
56	#14523977	—	2	1	no	—
57	#14523977	—	2	1	no	—
58	#14523975	—	2	4	no	—
59	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
60	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
61	Dic₆₇⋊C₈	—	2	2	no	—
62	Dic₆₇⋊C₈	—	2	2	no	—
63	#14523968	—	2	4	no	—
64	#14523968	—	2	4	no	—
65	#14523968	—	2	2	no	—
66	#14523968	—	2	2	no	—
67	C₈×Dic₆₇	—	2	2	no	—
68	C₈×Dic₆₇	—	2	2	no	—
69	C₂×M₁₆×C₆₇	—	2	2	no	—
70	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₆₇	—	2	2	no	—
74	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
75	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
76	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₃₄	—	2	1	no	—
80	C₂×Dic₁₃₄	—	2	2	no	—
81	C₂×Dic₁₃₄	—	2	2	no	—
82	C₂×Dic₁₃₄	—	2	2	no	—
83	C₂×Dic₁₃₄	Q₁₆	1	4	yes	complemented
84	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
85	C₂²×Dic₆₇	—	2	1	no	—
86	C₂²×Dic₆₇	—	2	1	no	—
87	Dic₂₆₈	—	4	2	no	—
88	Dic₂₆₈	—	4	2	no	—
89	C₂×C₆₇⋊C₈	—	2	2	no	—
90	C₂×C₆₇⋊C₈	—	4	2	no	—
91	C₄⋊Dic₆₇	—	2	1	no	—
92	C₄⋊Dic₆₇	—	2	2	no	—
93	C₄⋊Dic₆₇	—	2	2	no	—
94	C₄⋊Dic₆₇	—	2	2	no	—
95	C₄⋊Dic₆₇	—	2	2	no	—
96	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
97	Dic₆₇⋊C₄	—	2	2	no	—
98	Dic₆₇⋊C₄	—	2	2	no	—
99	Dic₆₇⋊C₄	—	4	2	no	—
100	Dic₆₇⋊C₄	—	4	2	no	—
101	C₂²⋊Dic₆₇	—	4	2	no	—
102	C₄×Dic₆₇	—	2	2	no	—
103	C₄×Dic₆₇	—	2	2	no	—
104	C₄×Dic₆₇	—	4	2	no	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
108	M₁₆×C₆₇	—	4	4	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	4	no	—
111	C₂×C₈×C₆₇	—	4	1	no	—
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₆₇	#41	1	1	yes	characteristic
119	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₂³×C₆₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	4	1	no	—
124	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
128	C₂×C₄×C₆₇	#135	1	2	yes	—
129	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
132	C₂×Dic₆₇	—	2	2	no	—
133	C₂×Dic₆₇	—	2	4	no	—
134	C₂×Dic₆₇	—	2	2	no	—
135	C₂×Dic₆₇	—	4	2	no	—
136	C₂×Dic₆₇	—	4	2	no	—
137	C₂×Dic₆₇	—	4	2	no	—
138	Dic₁₃₄	—	2	2	no	—
139	Dic₁₃₄	—	2	4	no	—
140	Dic₁₃₄	—	4	2	no	—
141	Dic₁₃₄	—	4	2	no	—
142	Dic₁₃₄	—	8	2	no	—
143	Dic₁₃₄	C₂×Q₁₆	1	4	yes	—
144	C₈×C₆₇	—	2	1	no	—
145	C₈×C₆₇	—	2	1	no	—
146	C₈×C₆₇	—	2	1	no	—
147	C₈×C₆₇	—	4	4	no	—
148	C₈×C₆₇	#137	1	2	yes	—
149	C₆₇⋊C₈	—	4	2	no	—
150	C₂²×C₆₇	—	2	1	no	—
151	C₂²×C₆₇	—	2	1	no	—
152	C₂²×C₆₇	#467	1	2	yes	—
153	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	#467	1	2	yes	—
158	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
159	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
160	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
161	C₄×C₆₇	#575	1	1	yes	characteristic
162	Dic₆₇	—	2	4	no	—
163	Dic₆₇	—	4	2	no	—
164	Dic₆₇	—	8	2	no	—
165	Dic₆₇	—	8	2	no	—
166	#19262	—	67	1	no	maximal, Sylow
167	C₂×C₆₇	—	2	2	no	—
168	C₂×C₆₇	#3037	1	1	yes	characteristic
169	C₂×C₆₇	C₂×#467	1	1	yes	characteristic
170	C₂×C₆₇	#3067	1	1	yes	characteristic
171	#3170	—	67	1	no	—
172	#3078	—	67	2	no	—
173	C₂×C₈×Q₈	—	67	1	no	—
174	#1684	—	67	4	no	—
175	#1676	—	67	2	no	—
176	#1670	—	67	2	no	—
177	#1567	D₆₇	1	1	yes	characteristic
178	#1550	—	67	2	no	—
179	C₆₇	#19262	1	1	yes	characteristic, complemented, Sylow
180	#533	—	67	2	no	—
181	C₂×C₄×Q₈	—	67	1	no	—
182	#500	—	67	1	no	—
183	#494	—	67	1	no	—
184	#487	—	67	2	no	—
185	#479	—	134	4	no	—
186	#477	—	67	4	no	—
187	#475	—	67	4	no	—
188	#467	—	134	1	no	—
189	#467	—	134	1	no	—
190	C₈×Q₈	—	67	2	no	—
191	C₈×Q₈	—	67	2	no	—
192	C₈×Q₈	—	134	1	no	—
193	C₈×Q₈	—	134	1	no	—
194	C₄×Q₁₆	—	134	1	no	—
195	C₄×Q₁₆	—	134	1	no	—
196	#428	—	134	2	no	—
197	#423	C₂×D₆₇	1	2	yes	—
198	C₂×C₄⋊C₈	—	67	1	no	—
199	C₂×C₄⋊C₈	—	134	1	no	—
200	#419	—	134	2	no	—
201	C₂×C₄×C₈	—	134	1	no	—
202	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
203	#335	C₂×D₆₇	1	2	yes	—
204	#335	C₂×D₆₇	1	2	yes	—
205	#333	—	67	4	no	—
206	#332	—	67	4	no	—
207	C₂²×Q₈	—	67	1	no	—
208	C₂×Q₁₆	—	134	1	no	—
209	C₂×Q₁₆	—	134	1	no	—
210	C₂×M₁₆	—	2	2	no	—
211	C₂²×C₈	—	134	1	no	—
212	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
213	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
214	C₄⋊Q₈	—	67	2	no	—
215	C₄⋊Q₈	—	67	2	no	—
216	#125	—	134	2	no	—
217	C₂²⋊Q₈	—	134	2	no	—
218	C₂²⋊Q₈	—	134	2	no	—
219	C₄×Q₈	—	67	2	no	—
220	C₄×Q₈	—	67	2	no	—
221	C₄×Q₈	—	134	2	no	—
222	C₄×Q₈	—	134	2	no	—
223	C₄×Q₈	—	134	1	no	—
224	C₄×Q₈	—	134	1	no	—
225	#117	—	134	2	no	—
226	C₂×C₄⋊C₄	—	67	1	no	—
227	C₂×C₄⋊C₄	—	134	1	no	—
228	C₂×C₄²	—	134	1	no	—
229	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
230	#107	—	134	4	no	—
231	#107	—	134	1	no	—
232	#107	—	134	1	no	—
233	#106	—	134	4	no	—
234	C₄⋊C₈	—	2	2	no	—
235	C₄⋊C₈	—	67	2	no	—
236	C₄⋊C₈	—	134	2	no	—
237	C₄⋊C₈	—	134	2	no	—
238	C₄⋊C₈	—	134	1	no	—
239	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
240	#103	—	134	4	no	—
241	#103	—	134	4	no	—
242	#103	—	134	2	no	—
243	#103	—	134	2	no	—
244	C₄×C₈	—	134	2	no	—
245	C₄×C₈	—	134	2	no	—
246	C₄×C₈	D₂₆₈	1	1	yes	characteristic
247	C₄×C₈	D₂₆₈	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
250	C₂×Q₈	—	67	4	no	—
251	C₂×Q₈	—	67	2	no	—
252	C₂×Q₈	—	134	2	no	—
253	C₂×Q₈	—	134	2	no	—
254	C₂×Q₈	—	134	1	no	—
255	C₂×Q₈	—	134	2	no	—
256	C₂²×C₄	—	134	1	no	—
257	C₂²×C₄	—	134	1	no	—
258	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
260	C₂²×C₄	#11761629	1	1	yes	characteristic
261	Q₁₆	—	268	2	no	—
262	Q₁₆	—	268	2	no	—
263	M₁₆	—	4	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	268	2	no	—
268	C₂×C₈	—	134	2	no	—
269	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
270	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
271	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
272	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
273	C₄⋊C₄	—	268	2	no	—
274	C₄⋊C₄	—	268	2	no	—
275	C₄⋊C₄	—	67	2	no	—
276	C₄⋊C₄	—	134	2	no	—
277	C₄⋊C₄	—	134	2	no	—
278	C₄⋊C₄	—	134	2	no	—
279	C₄⋊C₄	—	134	2	no	—
280	C₄⋊C₄	—	134	1	no	—
281	C₄⋊C₄	—	134	2	no	—
282	C₄⋊C₄	—	134	2	no	—
283	C₂²⋊C₄	—	268	2	no	—
284	C₄²	—	268	2	no	—
285	C₄²	—	134	2	no	—
286	C₄²	—	134	2	no	—
287	C₄²	D₄×D₆₇	1	1	yes	characteristic
288	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
289	C₄²	#11761629	1	1	yes	characteristic
290	C₄²	C₂³×D₆₇	1	1	yes	characteristic
291	C₂³	#14524018	1	1	yes	characteristic
292	Q₈	—	268	2	no	—
293	Q₈	—	268	2	no	—
294	Q₈	—	536	2	no	—
295	Q₈	—	67	4	no	—
296	Q₈	—	134	4	no	—
297	Q₈	—	134	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	268	2	no	—
302	C₂×C₄	—	268	2	no	—
303	C₂×C₄	—	268	2	no	—
304	C₂×C₄	—	134	2	no	—
305	C₂×C₄	—	134	2	no	—
306	C₂×C₄	—	134	4	no	—
307	C₂×C₄	#14524018	1	1	yes	characteristic
308	C₂×C₄	#14524018	1	1	yes	characteristic
309	C₂×C₄	#14524018	1	1	yes	characteristic
310	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
311	C₂×C₄	#14524062	1	2	yes	—
312	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
314	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	4	4	no	—
319	C₈	—	268	2	no	—
320	C₈	#14524076	1	2	yes	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	#16863638	1	2	yes	—
324	C₂²	C₂×#14524018	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	268	2	no	—
329	C₄	—	536	2	no	—
330	C₄	—	536	2	no	—
331	C₄	—	134	4	no	—
332	C₄	#16863638	1	2	yes	—
333	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
334	C₄	C₂×#14524062	1	1	yes	characteristic
335	C₄	C₂×#14524076	1	1	yes	characteristic
336	C₄	#16864196	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	#22569683	1	1	yes	characteristic
339	C₂	C₂×#16863638	1	1	yes	characteristic
340	C₂	#22569930	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524018
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1567	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂³×C₆₇	#121
hypercenter	#1567	D₆₇
hypocenter	C₆₇	#19262
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
4. #22559668	C₂	semidirect
5. #22559614	C₂	semidirect
6. #22559580	C₂	semidirect
83. C₂×Dic₁₃₄	Q₁₆	semidirect
179. C₆₇	#19262	semidirect

Maximal subgroups

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

subgroup	index
#22570795	2
#22570016	2
C₂×C₈×Dic₁₃₄	2
#22559668	2
#22559614	2
#22559580	2
#22558900	2
C₆₇×#1567	2
#19262	67