Subgroups — #30774330

Order 17152.

472 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863217	C₂	1	1	yes	maximal, characteristic, complemented
2	#22561945	C₂	1	2	yes	maximal, complemented
3	#22561937	C₂	1	2	yes	maximal, complemented
4	#22560088	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523979	C₂²	1	1	yes	characteristic
8	#16863434	C₂²	1	1	yes	characteristic
9	#16863351	—	2	1	no	—
10	#16863217	—	2	1	no	—
11	#16863217	—	2	1	no	—
12	#16863206	—	2	4	no	—
13	#16863203	—	2	4	no	—
14	#16863156	—	2	1	no	—
15	C₂×#135×C₆₇	—	2	1	no	—
16	C₂×#135×C₆₇	C₂²	1	2	yes	—
17	#532×C₆₇	—	2	1	no	—
18	#495×C₆₇	—	2	1	no	—
19	#495×C₆₇	C₄	1	2	yes	complemented
20	#494×C₆₇	C₄	1	1	yes	characteristic, complemented
21	#493×C₆₇	C₄	1	1	yes	characteristic, complemented
22	#492×C₆₇	—	2	1	no	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#11761612	—	2	1	no	—
25	#14524002	—	2	1	no	—
26	#14523983	—	2	2	no	—
27	#14523983	—	2	2	no	—
28	C₂×C₆₇⋊C₁₆	—	2	2	no	—
29	#14523979	—	2	1	no	—
30	#14523979	D₄	1	2	yes	complemented
31	#14523977	—	2	1	no	—
32	#14523975	D₄	1	2	yes	complemented
33	#14523953	—	4	1	no	—
34	C₈×Dic₆₇	—	4	1	no	—
35	C₂×#41×C₆₇	—	2	2	no	—
36	#135×C₆₇	—	2	2	no	—
37	#135×C₆₇	—	2	2	no	—
38	#135×C₆₇	—	2	2	no	—
39	#135×C₆₇	—	2	2	no	—
40	#135×C₆₇	—	4	1	no	—
41	#135×C₆₇	—	4	1	no	—
42	#135×C₆₇	—	4	1	no	—
43	#135×C₆₇	—	4	1	no	—
44	C₂×Q₁₆×C₆₇	—	2	1	no	—
45	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×SD₁₆×C₆₇	—	2	2	no	—
47	C₂×SD₁₆×C₆₇	—	2	2	no	—
48	C₂×D₈×C₆₇	—	2	1	no	—
49	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈×C₆₇	—	2	1	no	—
53	#127×C₆₇	—	2	1	no	—
54	#124×C₆₇	—	2	2	no	—
55	#121×C₆₇	—	4	2	no	—
56	C₄×D₄×C₆₇	—	4	2	no	—
57	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
62	#11761612	—	2	2	no	—
63	#11761612	—	4	1	no	—
64	C₂×C₆₇⋊C₈	—	4	1	no	—
65	C₄⋊Dic₆₇	—	2	2	no	—
66	C₂²⋊Dic₆₇	—	4	1	no	—
67	C₄×Dic₆₇	—	4	1	no	—
68	C₆₇⋊C₁₆	—	4	2	no	—
69	#41×C₆₇	—	4	2	no	—
70	#41×C₆₇	—	4	2	no	—
71	#41×C₆₇	—	4	2	no	—
72	#41×C₆₇	—	4	2	no	—
73	C₂×Q₈×C₆₇	—	2	2	no	—
74	C₂×D₄×C₆₇	—	2	2	no	—
75	C₂×D₄×C₆₇	—	4	2	no	—
76	C₂×D₄×C₆₇	—	4	2	no	—
77	C₂²×C₄×C₆₇	—	2	1	no	—
78	C₂²×C₄×C₆₇	—	4	2	no	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	Q₁₆×C₆₇	—	2	2	no	—
81	Q₁₆×C₆₇	—	2	2	no	—
82	Q₁₆×C₆₇	—	4	1	no	—
83	Q₁₆×C₆₇	—	4	1	no	—
84	SD₁₆×C₆₇	—	4	2	no	—
85	SD₁₆×C₆₇	—	4	2	no	—
86	SD₁₆×C₆₇	—	4	2	no	—
87	SD₁₆×C₆₇	—	4	2	no	—
88	D₈×C₆₇	—	2	2	no	—
89	D₈×C₆₇	—	2	2	no	—
90	D₈×C₆₇	—	4	1	no	—
91	D₈×C₆₇	—	4	1	no	—
92	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
95	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
96	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
97	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
103	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
104	C₄⋊C₄×C₆₇	—	4	2	no	—
105	C₂²⋊C₄×C₆₇	—	4	4	no	—
106	C₄²×C₆₇	—	2	1	no	—
107	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
108	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₆₇	—	4	2	no	—
110	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
111	Q₈×C₆₇	—	4	2	no	—
112	Q₈×C₆₇	—	4	2	no	—
113	D₄×C₆₇	—	4	2	no	—
114	D₄×C₆₇	—	4	2	no	—
115	D₄×C₆₇	—	8	4	no	—
116	D₄×C₆₇	—	8	2	no	—
117	D₄×C₆₇	—	8	2	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	8	2	no	—
123	C₂×C₄×C₆₇	—	8	2	no	—
124	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	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×D₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×Dic₆₇	—	4	2	no	—
131	C₈×C₆₇	#102	1	1	yes	characteristic
132	C₈×C₆₇	#102	1	1	yes	characteristic
133	C₈×C₆₇	#102	1	1	yes	characteristic
134	C₈×C₆₇	#102	1	1	yes	characteristic
135	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
136	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
137	C₈×C₆₇	#124	1	1	yes	characteristic
138	C₈×C₆₇	#127	1	1	yes	characteristic
139	C₆₇⋊C₈	—	4	2	no	—
140	C₂²×C₆₇	—	4	1	no	—
141	C₂²×C₆₇	—	8	4	no	—
142	C₂²×C₆₇	—	8	2	no	—
143	C₂²×C₆₇	#418	1	1	yes	characteristic
144	C₂²×C₆₇	#459	1	1	yes	characteristic
145	C₂²×C₆₇	#460	1	1	yes	characteristic
146	C₄×C₆₇	—	2	2	no	—
147	C₄×C₆₇	—	4	1	no	—
148	C₄×C₆₇	—	8	2	no	—
149	C₄×C₆₇	#390	1	1	yes	characteristic
150	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
151	C₄×C₆₇	#459	1	1	yes	characteristic
152	C₄×C₆₇	#460	1	1	yes	characteristic
153	Dic₆₇	—	8	2	no	—
154	#13259	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	1	no	—
156	C₂×C₆₇	—	2	1	no	—
157	C₂×C₆₇	—	2	1	no	—
158	C₂×C₆₇	—	16	2	no	—
159	C₂×C₆₇	#1982	1	1	yes	characteristic
160	#3147	D₆₇	1	1	yes	characteristic
161	C₂×#364	—	67	1	no	—
162	#2145	—	67	2	no	—
163	#2143	—	67	2	no	—
164	#1772	—	67	1	no	—
165	C₆₇	#13259	1	1	yes	characteristic, complemented, Sylow
166	C₂×#135	—	2	1	no	—
167	C₂×#135	C₂×D₆₇	1	2	yes	—
168	#532	—	2	1	no	—
169	C₂×M₃₂	—	67	2	no	—
170	#495	—	2	1	no	—
171	#495	Dic₆₇	1	2	yes	complemented
172	#494	Dic₆₇	1	1	yes	characteristic, complemented
173	#493	Dic₆₇	1	1	yes	characteristic, complemented
174	#492	—	2	1	no	—
175	C₂×#108	—	67	1	no	—
176	#427	—	67	1	no	—
177	#405	—	134	1	no	—
178	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
179	#364	—	134	1	no	—
180	#364	—	134	1	no	—
181	#360	—	134	4	no	—
182	#359	—	134	4	no	—
183	#339	—	134	1	no	—
184	C₂×#41	—	2	2	no	—
185	#135	—	2	2	no	—
186	#135	—	2	2	no	—
187	#135	—	2	2	no	—
188	#135	—	2	2	no	—
189	#135	—	4	1	no	—
190	#135	—	4	1	no	—
191	#135	—	4	1	no	—
192	#135	—	4	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
195	C₂×SD₁₆	—	2	2	no	—
196	C₂×SD₁₆	—	2	2	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
199	C₂×M₁₆	—	134	1	no	—
200	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
201	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
202	C₄⋊Q₈	—	2	1	no	—
203	#127	—	2	1	no	—
204	#124	—	2	2	no	—
205	#121	—	4	2	no	—
206	C₄×D₄	—	4	2	no	—
207	#117	—	134	1	no	—
208	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
209	M₃₂	—	134	2	no	—
210	M₃₂	—	134	2	no	—
211	C₂×C₁₆	—	134	2	no	—
212	#108	—	67	2	no	—
213	#108	—	134	1	no	—
214	#107	—	134	1	no	—
215	#106	—	67	2	no	—
216	#97	—	268	1	no	—
217	C₄×C₈	—	268	1	no	—
218	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
219	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
220	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
221	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
222	#41	—	4	2	no	—
223	#41	—	4	2	no	—
224	#41	—	4	2	no	—
225	#41	—	4	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	4	1	no	—
236	Q₁₆	—	4	1	no	—
237	SD₁₆	—	4	2	no	—
238	SD₁₆	—	4	2	no	—
239	SD₁₆	—	4	2	no	—
240	SD₁₆	—	4	2	no	—
241	D₈	—	2	2	no	—
242	D₈	—	2	2	no	—
243	D₈	—	4	1	no	—
244	D₈	—	4	1	no	—
245	M₁₆	—	268	1	no	—
246	M₁₆	—	134	2	no	—
247	C₂×C₈	—	268	1	no	—
248	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
250	C₂×C₈	#11761613	1	1	yes	characteristic
251	C₂×C₈	#11761613	1	1	yes	characteristic
252	C₂×C₈	#11761616	1	1	yes	characteristic
253	C₂×C₈	#11761616	1	1	yes	characteristic
254	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
255	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
256	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
257	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
258	C₂×C₈	#11761627	1	1	yes	characteristic
259	C₂×C₈	#11761627	1	1	yes	characteristic
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	134	2	no	—
262	C₂²⋊C₄	—	4	4	no	—
263	C₂²⋊C₄	—	268	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	268	1	no	—
266	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
267	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
268	C₁₆	—	268	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	#14524069	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	2	no	—
275	D₄	—	8	4	no	—
276	D₄	—	8	2	no	—
277	D₄	—	8	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	—	268	2	no	—
285	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
286	C₂×C₄	#14524014	1	1	yes	characteristic
287	C₂×C₄	#14524014	1	1	yes	characteristic
288	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
289	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
290	C₂×C₄	#14524069	1	1	yes	characteristic
291	C₈	—	268	2	no	—
292	C₈	#14523964	1	1	yes	characteristic
293	C₈	#14523964	1	1	yes	characteristic
294	C₈	#14523964	1	1	yes	characteristic
295	C₈	#14523964	1	1	yes	characteristic
296	C₈	D₄×Dic₆₇	1	1	yes	characteristic
297	C₈	D₄×Dic₆₇	1	1	yes	characteristic
298	C₈	#14524030	1	1	yes	characteristic
299	C₈	#14524039	1	1	yes	characteristic
300	C₂²	—	4	1	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	8	2	no	—
303	C₂²	#16863401	1	1	yes	characteristic
304	C₂²	#16863588	1	1	yes	characteristic
305	C₂²	#16863593	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	536	2	no	—
310	C₄	#16863299	1	1	yes	characteristic
311	C₄	C₂×#14523964	1	1	yes	characteristic
312	C₄	#16863588	1	1	yes	characteristic
313	C₄	#16863593	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	#22561137	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16863588
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting, prime radical	C₆₇×#3147	C₂
socle	C₂×C₆₇	#1982
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#3147	D₆₇
hypocenter	C₆₇	#13259
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#16863217	C₂	semidirect
2. #22561945	C₂	semidirect
3. #22561937	C₂	semidirect
4. #22560088	C₂	semidirect
19. #495×C₆₇	C₄	semidirect
20. #494×C₆₇	C₄	semidirect
21. #493×C₆₇	C₄	semidirect
30. #14523979	D₄	semidirect
32. #14523975	D₄	semidirect
165. C₆₇	#13259	semidirect
171. #495	Dic₆₇	semidirect
172. #494	Dic₆₇	semidirect
173. #493	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863217	2
#22561945	2
#22561937	2
#22560088	2
C₆₇×#3147	2
#13259	67