Subgroups — #30864182

Order 17152.

618 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560933	C₂	1	2	yes	maximal
2	#22560775	C₂	1	4	yes	maximal
3	#22560329	C₂	1	1	yes	maximal, characteristic
4	#22560084	C₂	1	4	yes	maximal
5	#22560078	C₂	1	2	yes	maximal
6	#22559896	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#2402	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×#14523954	C₂²	1	2	yes	—
11	C₂×#14523954	C₂²	1	4	yes	—
12	C₂×#14523954	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	#16863151	C₂²	1	4	yes	—
16	#16863151	C₂²	1	4	yes	—
17	#16863151	C₂²	1	4	yes	—
18	#400×C₆₇	C₄	1	4	yes	—
19	#398×C₆₇	C₄	1	4	yes	—
20	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	#376×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
24	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
25	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
26	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
27	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
29	#14523954	—	2	4	no	—
30	#14523954	—	2	8	no	—
31	#14523954	—	2	2	no	—
32	C₄×C₆₇⋊C₈	—	2	4	no	—
33	C₄×C₆₇⋊C₈	Q₈	1	4	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	#95×C₆₇	C₂×C₄	1	4	yes	—
41	#95×C₆₇	C₂×C₄	1	2	yes	—
42	#95×C₆₇	C₂×C₄	1	4	yes	—
43	#95×C₆₇	C₂×C₄	1	4	yes	—
44	#95×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₆₇⋊C₈	—	2	2	no	—
46	C₂×C₆₇⋊C₈	—	2	2	no	—
47	C₂×C₆₇⋊C₈	—	2	2	no	—
48	C₂×C₆₇⋊C₈	—	2	4	no	—
49	C₂×C₆₇⋊C₈	—	2	4	no	—
50	C₂×C₆₇⋊C₈	—	2	4	no	—
51	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
52	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
53	C₂×C₆₇⋊C₈	#41	1	4	yes	—
54	C₂×C₆₇⋊C₈	#41	1	4	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	4	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	2	yes	—
59	C₂²×C₄×C₆₇	C₂⁴	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	C₄²×C₆₇	—	2	2	no	—
64	C₄²×C₆₇	—	2	4	no	—
65	C₄²×C₆₇	—	2	2	no	—
66	C₄²×C₆₇	—	2	4	no	—
67	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
68	C₄²×C₆₇	#41	1	2	yes	—
69	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	—	2	4	no	—
73	C₂×C₄×C₆₇	—	2	4	no	—
74	C₂×C₄×C₆₇	—	2	4	no	—
75	C₂×C₄×C₆₇	—	2	4	no	—
76	C₂×C₄×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	—	2	4	no	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
81	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
84	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
85	C₆₇⋊C₈	—	2	4	no	—
86	C₆₇⋊C₈	—	2	2	no	—
87	C₆₇⋊C₈	—	4	4	no	—
88	C₆₇⋊C₈	#125	1	4	yes	—
89	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
90	C₂²×C₆₇	C₄×#41	1	2	yes	—
91	C₂²×C₆₇	#568	1	4	yes	—
92	C₄×C₆₇	—	2	4	no	—
93	C₄×C₆₇	—	4	4	no	—
94	C₄×C₆₇	—	4	4	no	—
95	C₄×C₆₇	—	4	4	no	—
96	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
97	C₄×C₆₇	#529	1	2	yes	—
98	C₄×C₆₇	#533	1	1	yes	characteristic
99	#21677	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	C₄×#125	1	1	yes	characteristic
101	C₂×C₆₇	#2961	1	2	yes	—
102	C₂×C₆₇	#2967	1	4	yes	—
103	#2402	D₆₇	1	1	yes	characteristic
104	#1940	—	67	2	no	—
105	#1917	—	67	4	no	—
106	#1832	—	67	1	no	—
107	#1771	—	67	4	no	—
108	#1770	—	67	2	no	—
109	#1726	—	67	1	no	—
110	C₆₇	#21677	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₄⋊C₈	—	67	2	no	—
112	C₂×C₄⋊C₈	—	67	4	no	—
113	C₂×#97	—	67	2	no	—
114	C₂×#97	—	67	1	no	—
115	C₂×#97	—	67	4	no	—
116	C₂×C₄×C₈	—	67	1	no	—
117	C₂×C₄×C₈	—	67	2	no	—
118	#400	Dic₆₇	1	4	yes	—
119	#398	Dic₆₇	1	4	yes	—
120	C₄×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
121	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
122	#376	C₂×D₆₇	1	1	yes	characteristic
123	#336	—	67	4	no	—
124	#336	—	67	4	no	—
125	#336	—	67	4	no	—
126	C₂²×C₈	—	67	1	no	—
127	C₂²×C₈	—	67	1	no	—
128	C₂²×C₈	—	67	2	no	—
129	C₂²×C₈	—	67	4	no	—
130	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
131	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
132	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
133	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
134	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
135	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
136	C₄⋊C₈	—	134	8	no	—
137	C₄⋊C₈	—	134	4	no	—
138	#97	—	134	8	no	—
139	#97	—	134	4	no	—
140	#97	—	134	2	no	—
141	C₄×C₈	—	67	4	no	—
142	C₄×C₈	—	134	4	no	—
143	#95	C₂×Dic₆₇	1	4	yes	—
144	#95	C₂×Dic₆₇	1	2	yes	—
145	#95	C₂×Dic₆₇	1	4	yes	—
146	#95	C₂×Dic₆₇	1	4	yes	—
147	#95	C₂×Dic₆₇	1	2	yes	—
148	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
149	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
150	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
151	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
152	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
153	C₂×C₈	—	67	2	no	—
154	C₂×C₈	—	67	4	no	—
155	C₂×C₈	—	67	2	no	—
156	C₂×C₈	—	67	4	no	—
157	C₂×C₈	—	134	4	no	—
158	C₂×C₈	—	134	4	no	—
159	C₂×C₈	—	134	4	no	—
160	C₂×C₈	—	134	2	no	—
161	C₂×C₈	—	134	2	no	—
162	C₂×C₈	—	134	2	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	4	no	—
170	C₄²	Q₈×D₆₇	1	2	yes	—
171	C₄²	#11761629	1	2	yes	—
172	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
184	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
185	C₂×C₄	#41×D₆₇	1	2	yes	—
186	C₂×C₄	#41×D₆₇	1	2	yes	—
187	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
188	C₈	—	268	4	no	—
189	C₈	—	67	4	no	—
190	C₈	—	134	4	no	—
191	C₈	—	134	2	no	—
192	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
193	C₂²	#41×Dic₆₇	1	2	yes	—
194	C₂²	#16864163	1	4	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	4	4	no	—
197	C₄	—	4	4	no	—
198	C₄	—	4	4	no	—
199	C₄	#125×D₆₇	1	1	yes	characteristic
200	C₄	#16863931	1	2	yes	—
201	C₄	#16863952	1	1	yes	characteristic
202	C₂	#125×Dic₆₇	1	1	yes	characteristic
203	C₂	#22569150	1	2	yes	—
204	C₂	#22569196	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#2402	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2402	D₆₇
hypocenter	C₆₇	#21677
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2402

Complements

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

N	H ≅ G/N	type
110. C₆₇	#21677	semidirect

Maximal subgroups

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

subgroup	index
#22560933	2
#22560775	2
#22560329	2
#22560084	2
#22560078	2
#22559896	2
C₆₇×#2402	2
#21677	67