Subgroups — #31011346

Order 17152.

474 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570746	C₂	1	2	yes	maximal
2	#22562496	C₂	1	2	yes	maximal, complemented
3	#22562493	C₂	1	2	yes	maximal, complemented
4	#22562491	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562490	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562487	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561994	C₂	1	2	yes	maximal
8	#22561992	C₂	1	1	yes	maximal, characteristic
9	#367×D₆₇	C₂	1	1	yes	maximal, characteristic
10	#22561804	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2253	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864026	—	2	2	no	—
13	C₂×#14523982	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
15	#16863806	—	2	2	no	—
16	#16863803	—	2	2	no	—
17	#16863801	C₂²	1	2	yes	—
18	#16863800	C₂²	1	2	yes	—
19	#16863797	C₂²	1	2	yes	—
20	#16863435	—	2	2	no	—
21	#16863433	C₂²	1	2	yes	—
22	#107×D₆₇	C₂²	1	2	yes	—
23	#16863350	C₂²	1	1	yes	characteristic
24	#16863224	C₂²	1	1	yes	characteristic
25	#16863224	C₂²	1	1	yes	characteristic
26	#16863224	C₂²	1	2	yes	—
27	#16863223	C₂²	1	2	yes	—
28	#16863223	C₂²	1	2	yes	—
29	#16863221	C₂²	1	2	yes	—
30	#16863221	C₂²	1	2	yes	—
31	#16863172	C₂²	1	1	yes	characteristic
32	#16863171	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
34	#499×C₆₇	C₂²	1	2	yes	—
35	#367×C₆₇	C₂²	1	1	yes	characteristic
36	#367×C₆₇	C₂²	1	1	yes	characteristic
37	#366×C₆₇	C₂²	1	2	yes	—
38	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#11761611	D₄	1	1	yes	characteristic
40	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
41	#14524043	—	2	2	no	—
42	#14524042	—	2	2	no	—
43	#14524042	—	4	2	no	—
44	#14524034	—	2	2	no	—
45	#14524033	—	4	2	no	—
46	#14524023	—	4	2	no	—
47	#14524021	—	4	2	no	—
48	#14524001	—	2	2	no	—
49	#14524000	D₄	1	1	yes	characteristic
50	C₄⋊C₄×D₆₇	—	2	2	no	—
51	#14523982	Q₈	1	4	yes	complemented
52	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
54	C₁₆×D₆₇	Q₈	1	4	yes	complemented
55	#14523977	—	2	2	no	—
56	#14523977	C₂³	1	2	yes	—
57	#14523977	C₂³	1	2	yes	—
58	#14523976	—	2	2	no	—
59	#14523976	—	2	2	no	—
60	#14523974	—	2	2	no	—
61	#14523974	—	2	2	no	—
62	#14523954	D₄	1	1	yes	characteristic
63	#14523953	D₄	1	1	yes	characteristic
64	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
65	#125×C₆₇	—	2	2	no	—
66	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
68	#107×C₆₇	C₂³	1	2	yes	—
69	#107×C₆₇	C₂³	1	2	yes	—
70	#106×C₆₇	—	2	2	no	—
71	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₃₄	—	4	2	no	—
73	C₂×Dic₁₃₄	—	4	2	no	—
74	C₂×C₄×D₆₇	—	4	2	no	—
75	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
76	#11761611	Q₁₆	1	4	yes	—
77	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
79	C₈×D₆₇	C₂×Q₈	1	2	yes	—
80	C₈×D₆₇	C₂×Q₈	1	2	yes	—
81	C₄⋊Dic₆₇	—	2	2	no	—
82	C₄⋊Dic₆₇	—	2	2	no	—
83	C₄⋊Dic₆₇	—	4	2	no	—
84	C₄⋊Dic₆₇	—	4	2	no	—
85	Dic₆₇⋊C₄	—	4	2	no	—
86	Dic₆₇⋊C₄	—	4	2	no	—
87	Dic₆₇⋊C₄	—	4	2	no	—
88	Dic₆₇⋊C₄	D₈	1	2	yes	—
89	(C₂×D₆₇)⋊C₄	—	4	2	no	—
90	(C₂×D₆₇)⋊C₄	D₈	1	2	yes	—
91	C₄×Dic₆₇	—	4	2	no	—
92	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
93	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
94	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₄⋊C₄×C₆₇	—	4	2	no	—
100	C₄⋊C₄×C₆₇	—	4	2	no	—
101	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
103	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
104	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×Dic₆₇	—	4	2	no	—
110	C₂×Dic₆₇	—	4	2	no	—
111	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
112	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
113	C₄×D₆₇	—	16	2	no	—
114	C₄×D₆₇	C₄⋊Q₈	1	2	yes	—
115	C₄×D₆₇	C₂×Q₁₆	1	2	yes	—
116	Dic₁₃₄	—	16	2	no	—
117	Dic₁₃₄	—	16	2	no	—
118	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
119	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
120	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
121	C₆₇⋊C₈	C₄⋊Q₈	1	2	yes	—
122	C₆₇⋊C₈	C₂×Q₁₆	1	2	yes	—
123	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
124	C₂×D₆₇	—	2	2	no	—
125	C₂×D₆₇	#500	1	2	yes	—
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	8	2	no	—
128	C₄×C₆₇	—	8	2	no	—
129	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
131	Dic₆₇	—	2	2	no	—
132	Dic₆₇	—	8	2	no	—
133	Dic₆₇	—	8	2	no	—
134	Dic₆₇	#500	1	2	yes	—
135	#33730	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
137	C₂×C₆₇	#3416	1	2	yes	—
138	D₆₇	—	2	2	no	—
139	#3164	—	67	2	no	—
140	#2255	—	67	2	no	—
141	#2255	—	67	2	no	—
142	#2254	—	67	1	no	—
143	#2254	—	67	1	no	—
144	#2253	—	67	1	no	—
145	#2253	D₆₇	1	1	yes	characteristic, complemented
146	#2159	—	67	2	no	—
147	#2158	—	67	1	no	—
148	C₂×#367	—	67	1	no	—
149	#2108	—	67	1	no	—
150	C₆₇	#33730	1	1	yes	characteristic, complemented, Sylow
151	#544	—	134	2	no	—
152	C₂×M₃₂	—	67	1	no	—
153	C₂²×C₁₆	—	67	1	no	—
154	#501	—	134	2	no	—
155	#501	—	134	2	no	—
156	#500	—	67	2	no	—
157	#500	—	67	2	no	—
158	#499	—	67	2	no	—
159	#499	C₂×D₆₇	1	2	yes	—
160	#428	—	134	2	no	—
161	#427	—	67	2	no	—
162	C₂×#107	—	67	2	no	—
163	#405	—	67	1	no	—
164	#367	—	67	1	no	—
165	#367	—	67	1	no	—
166	#367	—	67	2	no	—
167	#367	—	67	2	no	—
168	#367	—	67	2	no	—
169	#367	C₂×D₆₇	1	1	yes	characteristic
170	#367	C₂×D₆₇	1	1	yes	characteristic
171	#366	—	67	2	no	—
172	#366	—	67	2	no	—
173	#366	C₂×D₆₇	1	2	yes	—
174	#346	—	67	1	no	—
175	#346	—	67	1	no	—
176	C₄×C₁₆	—	67	1	no	—
177	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
178	C₂×M₁₆	—	67	1	no	—
179	C₂²×C₈	—	67	1	no	—
180	C₄⋊Q₈	—	268	2	no	—
181	C₄⋊Q₈	—	134	2	no	—
182	C₄⋊Q₈	—	134	2	no	—
183	#125	—	2	2	no	—
184	#125	—	268	2	no	—
185	#125	—	134	2	no	—
186	C₂²⋊Q₈	—	268	2	no	—
187	C₂²⋊Q₈	—	268	2	no	—
188	#117	—	67	1	no	—
189	#117	—	134	2	no	—
190	C₂×C₄⋊C₄	—	134	2	no	—
191	M₃₂	—	67	4	no	—
192	C₂×C₁₆	—	67	1	no	—
193	C₂×C₁₆	—	67	1	no	—
194	C₂×C₁₆	—	67	4	no	—
195	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
196	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
197	#107	—	67	2	no	—
198	#107	—	67	2	no	—
199	#107	—	134	2	no	—
200	#107	—	134	2	no	—
201	#107	—	134	2	no	—
202	#107	C₂²×D₆₇	1	2	yes	—
203	#107	C₂²×D₆₇	1	2	yes	—
204	#106	—	2	2	no	—
205	#106	—	134	2	no	—
206	#106	—	134	2	no	—
207	#97	—	67	1	no	—
208	#97	—	67	1	no	—
209	C₄×C₈	—	67	1	no	—
210	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
211	C₂×Q₈	—	268	2	no	—
212	C₂×Q₈	—	268	2	no	—
213	C₂²×C₄	—	268	2	no	—
214	C₂²×C₄	—	67	1	no	—
215	M₁₆	—	67	4	no	—
216	C₂×C₈	—	67	1	no	—
217	C₂×C₈	—	67	1	no	—
218	C₂×C₈	—	67	2	no	—
219	C₂×C₈	—	67	2	no	—
220	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
221	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	268	2	no	—
227	C₄⋊C₄	—	268	2	no	—
228	C₄⋊C₄	—	268	2	no	—
229	C₄⋊C₄	—	268	2	no	—
230	C₄⋊C₄	—	268	2	no	—
231	C₄⋊C₄	—	67	2	no	—
232	C₄⋊C₄	—	134	2	no	—
233	C₄⋊C₄	—	134	2	no	—
234	C₂²⋊C₄	—	268	2	no	—
235	C₂²⋊C₄	—	67	2	no	—
236	C₄²	—	268	2	no	—
237	C₄²	—	67	1	no	—
238	C₄²	D₄×D₆₇	1	1	yes	characteristic
239	C₁₆	—	67	2	no	—
240	C₁₆	—	67	2	no	—
241	C₁₆	Q₈×D₆₇	1	2	yes	—
242	C₁₆	Q₈×D₆₇	1	2	yes	—
243	C₂³	—	67	1	no	—
244	Q₈	—	1072	2	no	—
245	Q₈	—	1072	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	268	2	no	—
249	C₂×C₄	—	268	2	no	—
250	C₂×C₄	—	1072	2	no	—
251	C₂×C₄	—	67	1	no	—
252	C₂×C₄	—	67	2	no	—
253	C₂×C₄	—	67	2	no	—
254	C₂×C₄	—	67	2	no	—
255	C₂×C₄	D₈×D₆₇	1	2	yes	—
256	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
257	C₈	—	67	2	no	—
258	C₈	—	67	2	no	—
259	C₈	Q₁₆×D₆₇	1	2	yes	—
260	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
261	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
262	C₂²	—	67	2	no	—
263	C₂²	—	134	2	no	—
264	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	8	2	no	—
267	C₄	—	8	2	no	—
268	C₄	—	536	2	no	—
269	C₄	—	536	2	no	—
270	C₄	—	67	2	no	—
271	C₄	—	134	2	no	—
272	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
273	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
274	C₂	—	134	2	no	—
275	C₂	#500×D₆₇	1	1	yes	characteristic
276	C₂	#22573352	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2253	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂²×D₆₇	C₂×D₈
hypercenter	#2253	D₆₇
hypocenter	C₆₇	#33730
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
2. #22562496	C₂	semidirect
3. #22562493	C₂	semidirect
4. #22562491	C₂	semidirect
5. #22562490	C₂	semidirect
6. #22562487	C₂	semidirect
11. C₆₇×#2253	C₂	semidirect
51. #14523982	Q₈	semidirect
54. C₁₆×D₆₇	Q₈	semidirect
145. #2253	D₆₇	semidirect
150. C₆₇	#33730	semidirect

Maximal subgroups

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

subgroup	index
#22570746	2
#22562496	2
#22562493	2
#22562491	2
#22562490	2
#22562487	2
#22561994	2
#22561992	2
#367×D₆₇	2
#22561804	2
C₆₇×#2253	2
#33730	67