Subgroups — #30772185

Order 17152.

298 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863217	C₂	1	2	yes	maximal
2	#22558293	C₂	1	2	yes	maximal
3	#22558289	C₂	1	2	yes	maximal
4	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	4	yes	—
6	#16863217	—	2	4	no	—
7	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
8	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#376×C₆₇	C₄	1	1	yes	characteristic
14	#14523983	—	2	4	no	—
15	#14523983	—	2	4	no	—
16	C₂×C₆₇⋊C₁₆	—	2	4	no	—
17	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
24	#97×C₆₇	—	2	2	no	—
25	#97×C₆₇	—	2	2	no	—
26	#97×C₆₇	—	2	2	no	—
27	C₄×C₈×C₆₇	—	2	2	no	—
28	C₄×C₈×C₆₇	C₈	1	4	yes	—
29	C₄×C₈×C₆₇	D₄	1	2	yes	—
30	C₄×C₈×C₆₇	Q₈	1	2	yes	—
31	#95×C₆₇	—	2	2	no	—
32	C₆₇⋊C₁₆	—	4	4	no	—
33	C₂²×C₄×C₆₇	—	2	1	no	—
34	C₂²×C₄×C₆₇	—	2	1	no	—
35	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
36	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₇	—	2	1	no	—
38	C₂×C₈×C₆₇	—	2	1	no	—
39	C₂×C₈×C₆₇	—	2	1	no	—
40	C₂×C₈×C₆₇	—	2	1	no	—
41	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
45	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
46	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₆₇	#41	1	2	yes	—
49	C₂×C₈×C₆₇	#41	1	2	yes	—
50	C₄²×C₆₇	—	2	1	no	—
51	C₄²×C₆₇	—	2	1	no	—
52	C₄²×C₆₇	—	2	1	no	—
53	C₄²×C₆₇	—	2	1	no	—
54	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
55	C₄²×C₆₇	C₂×C₈	1	2	yes	—
56	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	4	2	no	—
61	C₂×C₄×C₆₇	—	4	2	no	—
62	C₂×C₄×C₆₇	—	4	2	no	—
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
69	C₈×C₆₇	—	2	2	no	—
70	C₈×C₆₇	—	2	2	no	—
71	C₈×C₆₇	C₄⋊C₈	1	4	yes	—
72	C₈×C₆₇	#124	1	2	yes	—
73	C₈×C₆₇	#125	1	2	yes	—
74	C₂²×C₆₇	—	2	2	no	—
75	C₂²×C₆₇	#423	1	1	yes	characteristic
76	C₂²×C₆₇	#431	1	1	yes	characteristic
77	C₂²×C₆₇	#432	1	1	yes	characteristic
78	C₄×C₆₇	—	4	2	no	—
79	C₄×C₆₇	—	4	2	no	—
80	C₄×C₆₇	—	4	2	no	—
81	C₄×C₆₇	#382	1	1	yes	characteristic
82	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₆₇	#431	1	1	yes	characteristic
84	C₄×C₆₇	#432	1	1	yes	characteristic
85	#12881	—	67	1	no	maximal, Sylow
86	C₂×C₆₇	—	2	1	no	—
87	C₂×C₆₇	—	2	1	no	—
88	C₂×C₆₇	#1831	1	1	yes	characteristic
89	C₂×C₆₇	#2166	1	2	yes	—
90	C₂×#364	—	67	2	no	—
91	#1726	D₆₇	1	1	yes	characteristic
92	#1377	—	67	2	no	—
93	#1375	—	67	2	no	—
94	C₆₇	#12881	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₃₂	—	67	4	no	—
96	C₂×#97	Dic₆₇	1	1	yes	characteristic
97	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
98	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
99	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
100	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
102	#376	Dic₆₇	1	1	yes	characteristic
103	#364	—	134	4	no	—
104	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
106	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
108	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
109	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
111	M₃₂	—	134	4	no	—
112	M₃₂	—	134	4	no	—
113	C₂×C₁₆	—	134	4	no	—
114	#97	—	2	2	no	—
115	#97	—	2	2	no	—
116	#97	—	2	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	C₆₇⋊C₈	1	4	yes	—
119	C₄×C₈	Dic₁₃₄	1	2	yes	—
120	C₄×C₈	D₂₆₈	1	2	yes	—
121	#95	—	2	2	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	#11761612	1	2	yes	—
125	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
131	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
132	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
133	C₂×C₈	#11761612	1	2	yes	—
134	C₂×C₈	#11761612	1	2	yes	—
135	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
136	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
137	C₂×C₈	#11761628	1	2	yes	—
138	C₂×C₈	#11761628	1	2	yes	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	C₄⋊Dic₆₇	1	2	yes	—
144	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
145	C₁₆	—	268	4	no	—
146	C₂³	C₂×#11761612	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
154	C₂×C₄	#14524002	1	1	yes	characteristic
155	C₂×C₄	#14524002	1	1	yes	characteristic
156	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
157	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
158	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	2	no	—
161	C₈	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
162	C₈	#14524031	1	2	yes	—
163	C₈	#14524035	1	2	yes	—
164	C₂²	—	2	2	no	—
165	C₂²	#16863420	1	1	yes	characteristic
166	C₂²	#16863449	1	1	yes	characteristic
167	C₂²	#16863453	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	#16863262	1	1	yes	characteristic
172	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
173	C₄	#16863449	1	1	yes	characteristic
174	C₄	#16863453	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	#22560326	1	1	yes	characteristic
178	C₂	#22562025	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524002
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1726	C₂
socle	C₂²×C₆₇	#432
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#12881
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726

Complements

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

N	H ≅ G/N	type
94. C₆₇	#12881	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863217	2
#22558293	2
#22558289	2
C₆₇×#1726	2
#12881	67