Subgroups — #30755987

Order 17152.

506 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560326	C₂	1	1	yes	maximal, characteristic
2	#22560062	C₂	1	1	yes	maximal, characteristic
3	#22558092	C₂	1	4	yes	maximal
4	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14523954	—	2	1	no	—
8	#16863151	—	2	1	no	—
9	#16863151	—	2	1	no	—
10	#16863151	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₄	1	2	yes	—
14	#384×C₆₇	—	2	1	no	—
15	#384×C₆₇	—	2	1	no	—
16	#384×C₆₇	C₄	1	2	yes	—
17	#384×C₆₇	C₂²	1	2	yes	—
18	C₄³×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₆₇⋊C₈	—	2	2	no	—
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
23	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	complemented
24	#14523954	—	4	2	no	—
25	C₂²×Q₈×C₆₇	—	2	2	no	—
26	C₄⋊Q₈×C₆₇	—	4	4	no	—
27	C₄⋊Q₈×C₆₇	—	4	4	no	—
28	C₄⋊Q₈×C₆₇	—	4	2	no	—
29	C₄⋊Q₈×C₆₇	—	4	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₆₇⋊C₈	—	2	4	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₂×Q₈×C₆₇	—	4	4	no	—
50	C₂×Q₈×C₆₇	—	4	4	no	—
51	C₂×Q₈×C₆₇	—	4	4	no	—
52	C₂²×C₄×C₆₇	—	2	4	no	—
53	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₆₇	—	4	4	no	—
61	C₄⋊C₄×C₆₇	—	4	4	no	—
62	C₄⋊C₄×C₆₇	—	4	4	no	—
63	C₄⋊C₄×C₆₇	—	4	4	no	—
64	C₄⋊C₄×C₆₇	—	4	2	no	—
65	C₄⋊C₄×C₆₇	—	4	2	no	—
66	C₄⋊C₄×C₆₇	—	4	4	no	—
67	C₄²×C₆₇	—	2	2	no	—
68	C₄²×C₆₇	—	2	2	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	2	2	no	—
72	C₄²×C₆₇	D₈	1	2	yes	—
73	C₄²×C₆₇	SD₁₆	1	2	yes	—
74	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
75	C₄²×C₆₇	#41	1	2	yes	—
76	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Q₈×C₆₇	—	8	8	no	—
78	C₂×C₄×C₆₇	—	2	1	no	—
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	4	4	no	—
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	—	4	4	no	—
91	C₂×C₄×C₆₇	#100	1	2	yes	—
92	C₂×C₄×C₆₇	#101	1	2	yes	—
93	C₂×C₄×C₆₇	#102	1	2	yes	—
94	C₂×C₄×C₆₇	#102	1	2	yes	—
95	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
102	C₂×C₄×C₆₇	#137	1	2	yes	—
103	C₂×C₄×C₆₇	#137	1	2	yes	—
104	C₆₇⋊C₈	—	4	4	no	—
105	C₆₇⋊C₈	—	8	2	no	—
106	C₂²×C₆₇	#391	1	1	yes	characteristic
107	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
108	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
109	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
110	C₂²×C₆₇	#418	1	1	yes	characteristic
111	C₂²×C₆₇	#419	1	1	yes	characteristic
112	C₂²×C₆₇	#419	1	1	yes	characteristic
113	C₄×C₆₇	—	2	2	no	—
114	C₄×C₆₇	—	2	2	no	—
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	—	8	4	no	—
119	C₄×C₆₇	#474	1	2	yes	—
120	C₄×C₆₇	#475	1	2	yes	—
121	C₄×C₆₇	#489	1	2	yes	—
122	C₄×C₆₇	#501	1	2	yes	—
123	#9905	—	67	1	no	maximal, Sylow
124	C₂×C₆₇	#1545	1	1	yes	characteristic
125	C₂×C₆₇	#1546	1	1	yes	characteristic
126	C₂×C₆₇	#1556	1	1	yes	characteristic
127	C₂×C₆₇	#1558	1	1	yes	characteristic
128	C₂×C₆₇	#1988	1	1	yes	characteristic
129	C₂×C₆₇	#1993	1	1	yes	characteristic
130	C₂×C₆₇	#1997	1	1	yes	characteristic
131	#2868	D₆₇	1	1	yes	characteristic
132	#1831	—	67	1	no	—
133	#1767	—	67	1	no	—
134	#1296	—	67	4	no	—
135	C₆₇	#9905	1	1	yes	characteristic, complemented, Sylow
136	C₂×C₄⋊Q₈	—	2	1	no	—
137	C₂×C₄⋊Q₈	—	2	1	no	—
138	C₂×C₄⋊Q₈	Dic₆₇	1	2	yes	—
139	C₂×C₄⋊C₈	—	67	2	no	—
140	C₂×C₄⋊C₈	—	67	2	no	—
141	C₂×#97	—	134	1	no	—
142	#384	—	2	1	no	—
143	#384	—	2	1	no	—
144	#384	Dic₆₇	1	2	yes	—
145	#384	C₂×D₆₇	1	2	yes	—
146	C₄³	C₂×D₆₇	1	1	yes	characteristic
147	#336	—	134	1	no	—
148	#336	—	134	1	no	—
149	#336	—	134	1	no	—
150	C₂²×Q₈	—	2	2	no	—
151	C₂²×C₈	—	134	2	no	—
152	C₂²×C₈	—	134	2	no	—
153	C₄⋊Q₈	—	4	4	no	—
154	C₄⋊Q₈	—	4	4	no	—
155	C₄⋊Q₈	—	4	2	no	—
156	C₄⋊Q₈	—	4	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
163	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
170	C₄⋊C₈	—	67	4	no	—
171	C₄⋊C₈	—	134	4	no	—
172	C₄⋊C₈	—	134	2	no	—
173	#97	—	268	2	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
182	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
185	C₂×C₈	—	268	2	no	—
186	C₂×C₈	—	268	2	no	—
187	C₂×C₈	—	268	2	no	—
188	C₂×C₈	—	268	2	no	—
189	C₂×C₈	—	268	2	no	—
190	C₂×C₈	—	134	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	#11761613	1	2	yes	—
204	C₄²	#11761616	1	2	yes	—
205	C₄²	Q₈×D₆₇	1	2	yes	—
206	C₄²	#11761629	1	2	yes	—
207	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
208	Q₈	—	8	8	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	#14523960	1	2	yes	—
223	C₂×C₄	#14523962	1	2	yes	—
224	C₂×C₄	#14523964	1	2	yes	—
225	C₂×C₄	#14523964	1	2	yes	—
226	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
227	C₂×C₄	#14524019	1	2	yes	—
228	C₂×C₄	#14524032	1	1	yes	characteristic
229	C₂×C₄	#14524041	1	1	yes	characteristic
230	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
231	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
232	C₂×C₄	#14524069	1	2	yes	—
233	C₂×C₄	#14524074	1	2	yes	—
234	C₂×C₄	#14524074	1	2	yes	—
235	C₈	—	268	4	no	—
236	C₈	—	536	2	no	—
237	C₂²	#16863303	1	1	yes	characteristic
238	C₂²	C₂×#14523960	1	1	yes	characteristic
239	C₂²	C₂×#14523962	1	1	yes	characteristic
240	C₂²	C₂×#14523964	1	1	yes	characteristic
241	C₂²	#16863401	1	1	yes	characteristic
242	C₂²	#16863405	1	1	yes	characteristic
243	C₂²	#16863405	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	8	4	no	—
250	C₄	#16863673	1	2	yes	—
251	C₄	#16863678	1	2	yes	—
252	C₄	#16863758	1	2	yes	—
253	C₄	#16863802	1	2	yes	—
254	C₂	#22558877	1	1	yes	characteristic
255	C₂	#22558881	1	1	yes	characteristic
256	C₂	#22558926	1	1	yes	characteristic
257	C₂	#22558935	1	1	yes	characteristic
258	C₂	#22561162	1	1	yes	characteristic
259	C₂	#22561183	1	1	yes	characteristic
260	C₂	#22561199	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2868	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2868	D₆₇
hypocenter	C₆₇	#9905
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
23. C₄⋊(C₆₇⋊C₈)	Q₈	semidirect
135. C₆₇	#9905	semidirect

Maximal subgroups

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

subgroup	index
#22560326	2
#22560062	2
#22558092	2
C₆₇×#2868	2
#9905	67