Subgroups — #30765924

Order 17152.

334 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#22558968	C₂	1	1	yes	maximal, characteristic
3	#22558173	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3170	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
6	#16863420	—	2	1	no	—
7	C₄×C₆₇⋊C₁₆	—	2	1	no	—
8	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	#16863149	—	2	1	no	—
10	#16863149	C₂²	1	2	yes	—
11	#533×C₆₇	—	2	1	no	—
12	#500×C₆₇	C₄	1	1	yes	characteristic
13	#494×C₆₇	C₄	1	1	yes	characteristic
14	#487×C₆₇	—	2	1	no	—
15	#467×C₆₇	—	2	1	no	—
16	#467×C₆₇	C₄	1	2	yes	—
17	C₄×Q₁₆×C₆₇	—	2	1	no	—
18	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#11761612	—	4	1	no	—
21	C₂×C₆₇⋊C₁₆	—	2	4	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
25	C₂×Q₁₆×C₆₇	—	2	1	no	—
26	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
29	C₄⋊Q₈×C₆₇	—	2	1	no	—
30	C₄⋊Q₈×C₆₇	—	2	1	no	—
31	#125×C₆₇	—	4	1	no	—
32	C₂²⋊Q₈×C₆₇	—	2	2	no	—
33	C₂²⋊Q₈×C₆₇	—	4	1	no	—
34	C₄×Q₈×C₆₇	—	2	2	no	—
35	C₄×Q₈×C₆₇	—	4	1	no	—
36	#117×C₆₇	—	4	1	no	—
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	#107×C₆₇	—	2	1	no	—
39	#107×C₆₇	C₂×C₄	1	2	yes	—
40	#103×C₆₇	—	2	2	no	—
41	#103×C₆₇	—	2	2	no	—
42	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	#11761612	—	8	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₆₇⋊C₁₆	—	2	4	no	—
49	C₂×Q₈×C₆₇	—	2	2	no	—
50	C₂×Q₈×C₆₇	—	4	1	no	—
51	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
54	Q₁₆×C₆₇	—	2	4	no	—
55	Q₁₆×C₆₇	—	4	2	no	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₄⋊C₄×C₆₇	—	4	1	no	—
64	C₄⋊C₄×C₆₇	—	4	1	no	—
65	C₄⋊C₄×C₆₇	—	4	2	no	—
66	C₄⋊C₄×C₆₇	—	4	1	no	—
67	C₄⋊C₄×C₆₇	—	4	2	no	—
68	C₂²⋊C₄×C₆₇	—	4	2	no	—
69	C₄²×C₆₇	—	4	2	no	—
70	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₆₇	D₈	1	1	yes	characteristic
72	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
73	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂³×C₆₇	#102	1	1	yes	characteristic
75	Q₈×C₆₇	—	4	2	no	—
76	Q₈×C₆₇	—	4	2	no	—
77	Q₈×C₆₇	—	8	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	4	2	no	—
82	C₂×C₄×C₆₇	—	4	2	no	—
83	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
84	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
86	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
87	C₂×C₄×C₆₇	Q₃₂	1	2	yes	—
88	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
91	C₈×C₆₇	—	2	1	no	—
92	C₈×C₆₇	—	2	1	no	—
93	C₈×C₆₇	#104	1	2	yes	—
94	C₈×C₆₇	#131	1	2	yes	—
95	C₆₇⋊C₈	—	8	2	no	—
96	C₂²×C₆₇	—	2	1	no	—
97	C₂²×C₆₇	—	2	1	no	—
98	C₂²×C₆₇	#358	1	2	yes	—
99	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
100	C₄×C₆₇	—	2	1	no	—
101	C₄×C₆₇	—	2	1	no	—
102	C₄×C₆₇	—	8	2	no	—
103	C₄×C₆₇	—	8	2	no	—
104	C₄×C₆₇	#358	1	2	yes	—
105	C₄×C₆₇	#359	1	2	yes	—
106	C₄×C₆₇	#408	1	1	yes	characteristic
107	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
108	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
109	C₄×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
110	#11672	—	67	1	no	maximal, Sylow
111	C₂×C₆₇	—	2	2	no	—
112	C₂×C₆₇	#1492	1	1	yes	characteristic
113	C₂×C₆₇	C₂×#358	1	1	yes	characteristic
114	C₂×C₆₇	C₂×#359	1	1	yes	characteristic
115	#3170	D₆₇	1	1	yes	characteristic
116	C₂×C₄×C₁₆	—	67	1	no	—
117	#1567	—	67	1	no	—
118	#1332	—	67	4	no	—
119	C₆₇	#11672	1	1	yes	characteristic, complemented, Sylow
120	#533	—	2	1	no	—
121	C₂²×C₁₆	—	134	1	no	—
122	#500	Dic₆₇	1	1	yes	characteristic
123	#494	Dic₆₇	1	1	yes	characteristic
124	#487	—	2	1	no	—
125	#467	—	2	1	no	—
126	#467	Dic₆₇	1	2	yes	—
127	C₄×Q₁₆	—	2	1	no	—
128	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
129	#423	—	134	1	no	—
130	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
131	C₄×C₁₆	—	67	2	no	—
132	C₄×C₁₆	—	134	1	no	—
133	#335	—	67	2	no	—
134	#335	—	134	1	no	—
135	C₂×Q₁₆	—	2	1	no	—
136	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
137	C₂×M₁₆	—	268	1	no	—
138	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
140	C₄⋊Q₈	—	2	1	no	—
141	C₄⋊Q₈	—	2	1	no	—
142	#125	—	4	1	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₂²⋊Q₈	—	4	1	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	4	1	no	—
147	#117	—	4	1	no	—
148	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
149	C₂×C₁₆	—	134	4	no	—
150	C₂×C₁₆	—	134	2	no	—
151	#107	—	2	1	no	—
152	#107	C₂×Dic₆₇	1	2	yes	—
153	C₄⋊C₈	—	268	1	no	—
154	C₄⋊C₈	—	134	2	no	—
155	#103	—	2	2	no	—
156	#103	—	2	2	no	—
157	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
164	C₂²×C₄	#11761613	1	1	yes	characteristic
165	C₂²×C₄	#11761616	1	1	yes	characteristic
166	Q₁₆	—	2	4	no	—
167	Q₁₆	—	4	2	no	—
168	M₁₆	—	536	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	268	2	no	—
172	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
174	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
185	C₄²	#11761613	1	1	yes	characteristic
186	C₄²	#11761616	1	1	yes	characteristic
187	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
188	C₁₆	—	134	4	no	—
189	C₂³	#14523964	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	8	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	#14523964	1	1	yes	characteristic
199	C₂×C₄	#14523964	1	1	yes	characteristic
200	C₂×C₄	#14523964	1	1	yes	characteristic
201	C₂×C₄	#14523986	1	2	yes	—
202	C₂×C₄	#14523989	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
204	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
205	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	536	2	no	—
209	C₈	#14523970	1	2	yes	—
210	C₈	#14524050	1	2	yes	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#16863200	1	2	yes	—
214	C₂²	C₂×#14523964	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	2	no	—
219	C₄	#16863200	1	2	yes	—
220	C₄	#16863203	1	2	yes	—
221	C₄	#16863363	1	1	yes	characteristic
222	C₄	C₂×#14523970	1	1	yes	characteristic
223	C₄	C₂×#14523986	1	1	yes	characteristic
224	C₄	C₂×#14523989	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	#22558660	1	1	yes	characteristic
227	C₂	C₂×#16863200	1	1	yes	characteristic
228	C₂	C₂×#16863203	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#3170	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	#102
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#3170	D₆₇
hypocenter	C₆₇	#11672
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #22558173	C₂	semidirect
119. C₆₇	#11672	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₆₇⋊C₁₆	2
#22558968	2
#22558173	2
C₆₇×#3170	2
#11672	67