Subgroups — #30755841

Order 17152.

494 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14524035	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22558048	C₂	1	4	yes	maximal
4	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524035	C₄	1	1	yes	characteristic
6	C₂×C₄×C₆₇⋊C₈	—	2	2	no	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
8	#16863294	C₄	1	2	yes	—
9	#16863280	—	2	2	no	—
10	#16863262	C₄	1	1	yes	characteristic
11	C₄×Dic₆₇⋊C₄	—	2	2	no	—
12	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₄³×C₆₇	C₂²	1	1	yes	characteristic
15	#336×C₆₇	—	2	2	no	—
16	C₂²×C₆₇⋊C₈	—	2	2	no	—
17	#14524035	—	2	2	no	—
18	#14524035	C₈	1	4	yes	complemented
19	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
21	C₂×Dic₆₇⋊C₄	—	2	2	no	—
22	C₂×Dic₆₇⋊C₄	—	2	2	no	—
23	C₂×C₄×Dic₆₇	—	2	2	no	—
24	C₄×C₆₇⋊C₈	—	2	8	no	—
25	C₄×C₆₇⋊C₈	—	2	2	no	—
26	C₄×C₆₇⋊C₈	Q₈	1	4	yes	—
27	C₆₇⋊#95	—	2	2	no	—
28	C₆₇⋊#95	—	2	2	no	—
29	C₆₇⋊#95	—	2	2	no	—
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	D₄	1	2	yes	—
34	C₂×C₄²×C₆₇	D₄	1	2	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₆₇	—	2	4	no	—
37	C₂²×Dic₆₇	—	2	2	no	—
38	C₂²×Dic₆₇	—	2	2	no	—
39	C₂×C₆₇⋊C₈	—	2	8	no	—
40	C₂×C₆₇⋊C₈	—	2	4	no	—
41	C₄⋊Dic₆₇	—	2	2	no	—
42	C₄⋊Dic₆₇	—	2	4	no	—
43	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
44	Dic₆₇⋊C₄	—	2	8	no	—
45	Dic₆₇⋊C₄	—	4	4	no	—
46	C₄×Dic₆₇	—	4	4	no	—
47	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	—	4	2	no	—
53	C₂×C₈×C₆₇	—	4	4	no	—
54	C₄²×C₆₇	—	2	4	no	—
55	C₄²×C₆₇	—	2	2	no	—
56	C₄²×C₆₇	—	2	4	no	—
57	C₄²×C₆₇	C₂×C₈	1	2	yes	—
58	C₄²×C₆₇	M₁₆	1	2	yes	—
59	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
60	C₄²×C₆₇	#41	1	2	yes	—
61	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	C₂²⋊C₈	1	4	yes	—
68	C₂×C₄×C₆₇	#104	1	4	yes	—
69	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#131	1	2	yes	—
76	C₂×C₄×C₆₇	#135	1	4	yes	—
77	C₂×Dic₆₇	—	2	4	no	—
78	C₂×Dic₆₇	—	4	2	no	—
79	C₂×Dic₆₇	—	4	2	no	—
80	C₂×Dic₆₇	—	4	2	no	—
81	C₂×Dic₆₇	—	4	2	no	—
82	C₂×Dic₆₇	—	4	2	no	—
83	C₈×C₆₇	—	8	2	no	—
84	C₆₇⋊C₈	—	2	8	no	—
85	C₂²×C₆₇	#391	1	1	yes	characteristic
86	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
87	C₂²×C₆₇	#408	1	1	yes	characteristic
88	C₂²×C₆₇	#416	1	2	yes	—
89	C₂²×C₆₇	C₂×#104	1	2	yes	—
90	C₄×C₆₇	—	2	4	no	—
91	C₄×C₆₇	—	2	2	no	—
92	C₄×C₆₇	—	2	4	no	—
93	C₄×C₆₇	C₈×Q₈	1	2	yes	—
94	C₄×C₆₇	#446	1	2	yes	—
95	C₄×C₆₇	#488	1	2	yes	—
96	C₄×C₆₇	#499	1	2	yes	—
97	Dic₆₇	—	4	4	no	—
98	Dic₆₇	—	8	2	no	—
99	#9871	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	#1476	1	2	yes	—
101	C₂×C₆₇	#1494	1	2	yes	—
102	C₂×C₆₇	#1984	1	1	yes	characteristic
103	C₂×C₆₇	#1991	1	1	yes	characteristic
104	C₂×C₆₇	#1994	1	1	yes	characteristic
105	C₄×#125	—	67	1	no	—
106	#1831	D₆₇	1	1	yes	characteristic
107	C₄²×C₈	—	67	1	no	—
108	#1279	—	67	4	no	—
109	C₆₇	#9871	1	1	yes	characteristic, complemented, Sylow
110	C₂×#125	—	67	1	no	—
111	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
112	C₂×C₄×C₈	—	67	2	no	—
113	C₂×C₄×C₈	—	134	2	no	—
114	#389	—	67	2	no	—
115	#387	—	134	2	no	—
116	#382	—	67	1	no	—
117	C₄×C₄⋊C₄	—	67	2	no	—
118	C₄×C₄⋊C₄	—	134	2	no	—
119	C₄³	C₂×D₆₇	1	1	yes	characteristic
120	#336	—	2	2	no	—
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	134	2	no	—
123	#125	—	67	4	no	—
124	#125	—	134	2	no	—
125	C₂×C₄⋊C₄	—	67	2	no	—
126	C₂×C₄⋊C₄	—	67	2	no	—
127	C₂×C₄⋊C₄	—	134	2	no	—
128	C₂×C₄⋊C₄	—	134	2	no	—
129	C₂×C₄²	—	134	2	no	—
130	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
132	C₂×C₄²	D₂₆₈	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
134	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
135	C₄⋊C₈	—	2	4	no	—
136	C₄×C₈	—	67	4	no	—
137	C₄×C₈	—	134	8	no	—
138	C₄×C₈	—	134	2	no	—
139	#95	—	134	2	no	—
140	#95	—	134	2	no	—
141	#95	—	134	2	no	—
142	C₂²×C₄	—	134	2	no	—
143	C₂²×C₄	—	134	2	no	—
144	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
145	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	4	no	—
151	C₂×C₈	—	134	8	no	—
152	C₂×C₈	—	134	4	no	—
153	C₄⋊C₄	—	268	4	no	—
154	C₄⋊C₄	—	67	4	no	—
155	C₄⋊C₄	—	134	4	no	—
156	C₄⋊C₄	—	134	8	no	—
157	C₄⋊C₄	—	134	2	no	—
158	C₄²	—	2	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	268	4	no	—
162	C₄²	C₈×D₆₇	1	2	yes	—
163	C₄²	#11761611	1	2	yes	—
164	C₄²	Q₈×D₆₇	1	2	yes	—
165	C₄²	#11761627	1	2	yes	—
166	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	268	2	no	—
173	C₂×C₄	—	268	2	no	—
174	C₂×C₄	—	268	2	no	—
175	C₂×C₄	—	268	2	no	—
176	C₂×C₄	—	268	2	no	—
177	C₂×C₄	—	134	4	no	—
178	C₂×C₄	(C₂×D₆₇)⋊C₈	1	4	yes	—
179	C₂×C₄	#14523969	1	4	yes	—
180	C₂×C₄	#14524008	1	2	yes	—
181	C₂×C₄	#14524023	1	2	yes	—
182	C₂×C₄	#14524030	1	1	yes	characteristic
183	C₂×C₄	#14524041	1	1	yes	characteristic
184	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
185	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
186	C₂×C₄	#14524051	1	2	yes	—
187	C₂×C₄	#14524063	1	4	yes	—
188	C₈	—	8	2	no	—
189	C₈	—	134	8	no	—
190	C₂²	#16863301	1	1	yes	characteristic
191	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
192	C₂²	#16863364	1	1	yes	characteristic
193	C₂²	#16863390	1	2	yes	—
194	C₂²	C₂×#14523969	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	4	no	—
198	C₄	—	268	4	no	—
199	C₄	—	536	2	no	—
200	C₄	#16863520	1	2	yes	—
201	C₄	#16863523	1	2	yes	—
202	C₄	#16863750	1	2	yes	—
203	C₄	#16863795	1	2	yes	—
204	C₂	#22558589	1	2	yes	—
205	C₂	#22558667	1	2	yes	—
206	C₂	#22561144	1	1	yes	characteristic
207	C₂	#22561173	1	1	yes	characteristic
208	C₂	#22561185	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1831	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1831	D₆₇
hypocenter	C₆₇	#9871
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
18. #14524035	C₈	semidirect
109. C₆₇	#9871	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14524035	2
C₄²×C₆₇⋊C₈	2
#22558048	2
C₆₇×#1831	2
#9871	67