Subgroups — #30869402

Order 17152.

722 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863527	C₂	1	2	yes	maximal
2	#22561146	C₂	1	2	yes	maximal
3	#22560775	C₂	1	8	yes	maximal
4	#22560326	C₂	1	1	yes	maximal, characteristic
5	#22559955	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863527	—	2	8	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×#14523954	C₂²	1	4	yes	—
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#16863151	C₂²	1	4	yes	—
14	#16863151	C₂²	1	8	yes	—
15	#16863151	C₂²	1	4	yes	—
16	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
18	#391×C₆₇	C₄	1	2	yes	—
19	#389×C₆₇	C₄	1	4	yes	—
20	#384×C₆₇	C₄	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
22	C₄³×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
24	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
25	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
26	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
28	#14523954	—	2	8	no	—
29	C₄×C₆₇⋊C₈	—	2	2	no	—
30	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
31	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₆₇	—	2	4	no	—
33	C₄×Q₈×C₆₇	—	2	4	no	—
34	C₄×Q₈×C₆₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	#95×C₆₇	C₂×C₄	1	8	yes	—
46	C₂×C₆₇⋊C₈	—	2	4	no	—
47	C₂×C₆₇⋊C₈	—	2	4	no	—
48	C₂×C₆₇⋊C₈	—	2	2	no	—
49	C₂×C₆₇⋊C₈	—	2	8	no	—
50	C₂×C₆₇⋊C₈	—	2	4	no	—
51	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
52	C₂×Q₈×C₆₇	—	2	4	no	—
53	C₂×Q₈×C₆₇	—	2	4	no	—
54	C₂×Q₈×C₆₇	—	2	4	no	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₄⋊C₄×C₆₇	—	2	8	no	—
64	C₄⋊C₄×C₆₇	—	2	8	no	—
65	C₄⋊C₄×C₆₇	—	4	4	no	—
66	C₄²×C₆₇	—	2	2	no	—
67	C₄²×C₆₇	—	2	4	no	—
68	C₄²×C₆₇	—	2	4	no	—
69	C₄²×C₆₇	—	2	4	no	—
70	C₄²×C₆₇	—	4	2	no	—
71	C₄²×C₆₇	M₁₆	1	4	yes	—
72	C₄²×C₆₇	C₂×D₄	1	2	yes	—
73	C₄²×C₆₇	#41	1	2	yes	—
74	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
75	Q₈×C₆₇	—	4	8	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	2	4	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	4	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	8	no	—
87	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
90	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₆₇	#131	1	2	yes	—
92	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
94	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
95	C₆₇⋊C₈	—	2	4	no	—
96	C₆₇⋊C₈	—	4	4	no	—
97	C₆₇⋊C₈	—	4	2	no	—
98	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
99	C₂²×C₆₇	#519	1	2	yes	—
100	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
101	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
102	C₂²×C₆₇	#568	1	2	yes	—
103	C₄×C₆₇	—	2	2	no	—
104	C₄×C₆₇	—	2	4	no	—
105	C₄×C₆₇	—	4	4	no	—
106	C₄×C₆₇	—	4	4	no	—
107	C₄×C₆₇	—	4	2	no	—
108	C₄×C₆₇	#436	1	4	yes	—
109	C₄×C₆₇	#536	1	2	yes	—
110	C₄×C₆₇	#541	1	2	yes	—
111	#22242	—	67	1	no	maximal, Sylow
112	C₂×C₆₇	#2327	1	1	yes	characteristic
113	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
114	C₂×C₆₇	#2932	1	1	yes	characteristic
115	C₂×C₆₇	#2983	1	2	yes	—
116	C₂×C₆₇	#2985	1	2	yes	—
117	C₂×#446	—	67	2	no	—
118	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
119	#1984	—	67	2	no	—
120	#1917	—	67	8	no	—
121	#1831	—	67	1	no	—
122	#1744	—	67	1	no	—
123	C₆₇	#22242	1	1	yes	characteristic, complemented, Sylow
124	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
125	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
126	#446	—	134	8	no	—
127	C₂×C₄⋊C₈	—	67	2	no	—
128	C₂×C₄⋊C₈	—	67	4	no	—
129	C₂×#97	—	67	4	no	—
130	C₂×C₄×C₈	—	67	1	no	—
131	C₂×C₄×C₈	—	67	1	no	—
132	#391	Dic₆₇	1	2	yes	—
133	#389	Dic₆₇	1	4	yes	—
134	#384	Dic₆₇	1	1	yes	characteristic
135	C₄×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
136	C₄³	C₂×D₆₇	1	1	yes	characteristic
137	#336	—	67	4	no	—
138	#336	—	67	8	no	—
139	#336	—	67	4	no	—
140	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
141	C₂²×C₈	—	67	2	no	—
142	C₂²×C₈	—	67	2	no	—
143	C₂²×C₈	—	67	4	no	—
144	C₄⋊Q₈	—	2	4	no	—
145	C₄×Q₈	—	2	4	no	—
146	C₄×Q₈	—	2	4	no	—
147	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
148	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
149	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
150	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
151	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
153	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
154	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
155	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
156	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
157	C₄⋊C₈	—	134	8	no	—
158	C₄⋊C₈	—	134	4	no	—
159	#97	—	134	8	no	—
160	C₄×C₈	—	67	4	no	—
161	C₄×C₈	—	134	2	no	—
162	#95	C₂×Dic₆₇	1	8	yes	—
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	2	4	no	—
166	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
167	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
168	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
169	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
171	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
172	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
173	C₂×C₈	—	67	4	no	—
174	C₂×C₈	—	134	4	no	—
175	C₂×C₈	—	134	2	no	—
176	C₂×C₈	—	134	4	no	—
177	C₂×C₈	—	134	8	no	—
178	C₂×C₈	—	134	4	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	8	no	—
181	C₄⋊C₄	—	2	8	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	4	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	2	no	—
188	C₄²	#11761612	1	4	yes	—
189	C₄²	D₄×D₆₇	1	2	yes	—
190	C₄²	#11761627	1	2	yes	—
191	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
192	Q₈	—	4	8	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
205	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
206	C₂×C₄	C₂×#11761612	1	4	yes	—
207	C₂×C₄	C₂×#11761612	1	2	yes	—
208	C₂×C₄	#14524050	1	2	yes	—
209	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
210	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
211	C₂×C₄	#14524091	1	4	yes	—
212	C₈	—	268	2	no	—
213	C₈	—	268	4	no	—
214	C₈	—	134	4	no	—
215	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
216	C₂²	#16863874	1	2	yes	—
217	C₂²	C₂²×#11761612	1	1	yes	characteristic
218	C₂²	C₂×#14524050	1	1	yes	characteristic
219	C₂²	#16864163	1	2	yes	—
220	C₄	—	2	4	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	4	no	—
224	C₄	—	4	4	no	—
225	C₄	#16863474	1	4	yes	—
226	C₄	#16863971	1	2	yes	—
227	C₄	#16864007	1	2	yes	—
228	C₂	#22562929	1	1	yes	characteristic
229	C₂	C₂×#16863474	1	1	yes	characteristic
230	C₂	#22568927	1	1	yes	characteristic
231	C₂	#22569294	1	2	yes	—
232	C₂	#22569304	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₄×C₄⋊Q₈×C₆₇	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#22242
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
123. C₆₇	#22242	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863527	2
#22561146	2
#22560775	2
#22560326	2
#22559955	2
C₄×C₄⋊Q₈×C₆₇	2
#22242	67