Subgroups — C₂²×#381×C₆₇ #30713256

Order 17152.

8062 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal
3	C₂×#381×C₆₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#95×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₆₇	—	2	16	no	—
7	C₂⁴×C₄×C₆₇	C₄	1	16	yes	complemented
8	C₂⁴×C₄×C₆₇	C₂²	1	4	yes	—
9	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₆₇	C₂²	1	12	yes	—
11	C₂²×C₂²⋊C₄×C₆₇	—	2	16	no	—
12	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	24	yes	—
13	#381×C₆₇	C₂²	1	64	yes	direct product factor
14	C₂×#95×C₆₇	C₂²	1	48	yes	complemented
15	C₂⁵×C₆₇	—	2	16	no	—
16	C₂⁵×C₆₇	C₂×C₄	1	8	yes	—
17	C₂⁵×C₆₇	D₄	1	16	yes	—
18	C₂⁵×C₆₇	C₂³	1	6	yes	—
19	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₆₇	—	2	128	no	—
21	C₂³×C₄×C₆₇	—	2	16	no	—
22	C₂³×C₄×C₆₇	—	2	96	no	—
23	C₂³×C₄×C₆₇	—	2	16	no	—
24	C₂³×C₄×C₆₇	—	2	32	no	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	16	yes	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	96	yes	complemented
27	C₂³×C₄×C₆₇	C₂³	1	24	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	12	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	16	yes	—
30	C₂×C₂²⋊C₄×C₆₇	—	2	96	no	—
31	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	—
32	#95×C₆₇	C₂³	1	64	yes	complemented
33	C₂⁴×C₆₇	—	2	64	no	—
34	C₂⁴×C₆₇	—	2	128	no	—
35	C₂⁴×C₆₇	—	2	48	no	—
36	C₂⁴×C₆₇	C₂²⋊C₄	1	64	yes	—
37	C₂⁴×C₆₇	C₂²×C₄	1	24	yes	—
38	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
39	C₂⁴×C₆₇	C₂×D₄	1	48	yes	—
40	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
41	C₂⁴×C₆₇	#41	1	16	yes	—
42	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
43	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
44	C₂²×C₄×C₆₇	—	2	64	no	—
45	C₂²×C₄×C₆₇	—	2	384	no	—
46	C₂²×C₄×C₆₇	—	2	48	no	—
47	C₂²×C₄×C₆₇	—	2	64	no	—
48	C₂²×C₄×C₆₇	—	2	48	no	—
49	C₂²×C₄×C₆₇	—	2	96	no	—
50	C₂²×C₄×C₆₇	—	4	64	no	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	48	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	64	yes	complemented
53	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
54	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
55	C₂²⋊C₄×C₆₇	—	2	64	no	—
56	C₂³×C₆₇	—	2	96	no	—
57	C₂³×C₆₇	—	2	192	no	—
58	C₂³×C₆₇	—	2	16	no	—
59	C₂³×C₆₇	—	2	256	no	—
60	C₂³×C₆₇	C₂×C₂²⋊C₄	1	96	yes	—
61	C₂³×C₆₇	C₂×C₂²⋊C₄	1	16	yes	—
62	C₂³×C₆₇	#117	1	32	yes	—
63	C₂³×C₆₇	#123	1	64	yes	—
64	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
65	C₂³×C₆₇	C₂³×C₄	1	6	yes	—
66	C₂³×C₆₇	C₂²×D₄	1	16	yes	—
67	C₂³×C₆₇	C₂²×D₄	1	12	yes	—
68	C₂³×C₆₇	C₂×#41	1	24	yes	—
69	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	96	no	—
71	C₂×C₄×C₆₇	—	2	128	no	—
72	C₂×C₄×C₆₇	—	2	16	no	—
73	C₂×C₄×C₆₇	—	2	16	no	—
74	C₂×C₄×C₆₇	—	2	32	no	—
75	C₂×C₄×C₆₇	—	4	96	no	—
76	C₂×C₄×C₆₇	C₂³×C₄	1	16	yes	—
77	C₂²×C₆₇	—	2	16	no	—
78	C₂²×C₆₇	—	2	32	no	—
79	C₂²×C₆₇	—	2	192	no	—
80	C₂²×C₆₇	#381	1	64	yes	direct product factor
81	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	16	yes	—
82	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	12	yes	—
83	C₂²×C₆₇	C₂×#117	1	24	yes	—
84	C₂²×C₆₇	C₂×#123	1	48	yes	—
85	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
86	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
87	C₂²×C₆₇	C₂²×#41	1	4	yes	—
88	C₄×C₆₇	—	2	16	no	—
89	C₄×C₆₇	—	4	16	no	—
90	C₂²×#381	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₆₇	—	2	16	no	—
92	C₂×C₆₇	C₂×#381	1	24	yes	direct product factor
93	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₆₇	C₂²×#117	1	2	yes	—
95	C₂×C₆₇	C₂²×#123	1	4	yes	—
96	C₂⁵×C₄	C₂×C₆₇	1	1	yes	characteristic
97	C₂³×C₂²⋊C₄	C₂×C₆₇	1	2	yes	—
98	C₂×#381	C₂×C₆₇	1	24	yes	direct product factor
99	C₂²×#95	C₂×C₆₇	1	4	yes	complemented
100	C₆₇	C₂²×#381	1	1	yes	characteristic, direct product factor, Sylow
101	C₂⁶	C₂²×C₆₇	1	1	yes	characteristic
102	C₂⁴×C₄	—	2	16	no	—
103	C₂⁴×C₄	C₄×C₆₇	1	16	yes	complemented
104	C₂⁴×C₄	C₂²×C₆₇	1	4	yes	—
105	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
106	C₂⁴×C₄	C₂²×C₆₇	1	12	yes	—
107	C₂²×C₂²⋊C₄	—	2	16	no	—
108	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	24	yes	—
109	#381	C₂²×C₆₇	1	64	yes	direct product factor
110	C₂×#95	C₂²×C₆₇	1	48	yes	complemented
111	C₂⁵	—	2	16	no	—
112	C₂⁵	C₂×C₄×C₆₇	1	8	yes	—
113	C₂⁵	D₄×C₆₇	1	16	yes	—
114	C₂⁵	C₂³×C₆₇	1	6	yes	—
115	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
116	C₂³×C₄	—	2	128	no	—
117	C₂³×C₄	—	2	16	no	—
118	C₂³×C₄	—	2	96	no	—
119	C₂³×C₄	—	2	16	no	—
120	C₂³×C₄	—	2	32	no	—
121	C₂³×C₄	C₂×C₄×C₆₇	1	16	yes	—
122	C₂³×C₄	C₂×C₄×C₆₇	1	96	yes	complemented
123	C₂³×C₄	C₂³×C₆₇	1	24	yes	—
124	C₂³×C₄	C₂³×C₆₇	1	12	yes	—
125	C₂³×C₄	C₂³×C₆₇	1	16	yes	—
126	C₂×C₂²⋊C₄	—	2	96	no	—
127	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	—
128	#95	C₂³×C₆₇	1	64	yes	complemented
129	C₂⁴	—	2	64	no	—
130	C₂⁴	—	2	128	no	—
131	C₂⁴	—	2	48	no	—
132	C₂⁴	C₂²⋊C₄×C₆₇	1	64	yes	—
133	C₂⁴	C₂²×C₄×C₆₇	1	24	yes	—
134	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
135	C₂⁴	C₂×D₄×C₆₇	1	48	yes	—
136	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
137	C₂⁴	#41×C₆₇	1	16	yes	—
138	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
139	C₂⁴	C₂⁴×C₆₇	1	3	yes	—
140	C₂²×C₄	—	2	64	no	—
141	C₂²×C₄	—	2	384	no	—
142	C₂²×C₄	—	2	48	no	—
143	C₂²×C₄	—	2	64	no	—
144	C₂²×C₄	—	2	48	no	—
145	C₂²×C₄	—	2	96	no	—
146	C₂²×C₄	—	4	64	no	—
147	C₂²×C₄	C₂²×C₄×C₆₇	1	48	yes	—
148	C₂²×C₄	C₂²×C₄×C₆₇	1	64	yes	complemented
149	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
150	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
151	C₂²⋊C₄	—	2	64	no	—
152	C₂³	—	2	96	no	—
153	C₂³	—	2	192	no	—
154	C₂³	—	2	16	no	—
155	C₂³	—	2	256	no	—
156	C₂³	C₂×C₂²⋊C₄×C₆₇	1	96	yes	—
157	C₂³	C₂×C₂²⋊C₄×C₆₇	1	16	yes	—
158	C₂³	#117×C₆₇	1	32	yes	—
159	C₂³	#123×C₆₇	1	64	yes	—
160	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
161	C₂³	C₂³×C₄×C₆₇	1	6	yes	—
162	C₂³	C₂²×D₄×C₆₇	1	16	yes	—
163	C₂³	C₂²×D₄×C₆₇	1	12	yes	—
164	C₂³	C₂×#41×C₆₇	1	24	yes	—
165	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
166	C₂×C₄	—	2	96	no	—
167	C₂×C₄	—	2	128	no	—
168	C₂×C₄	—	2	16	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	—	2	32	no	—
171	C₂×C₄	—	4	96	no	—
172	C₂×C₄	C₂³×C₄×C₆₇	1	16	yes	—
173	C₂²	—	2	192	no	—
174	C₂²	—	2	16	no	—
175	C₂²	—	2	32	no	—
176	C₂²	#381×C₆₇	1	64	yes	direct product factor
177	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	16	yes	—
178	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	12	yes	—
179	C₂²	C₂×#117×C₆₇	1	24	yes	—
180	C₂²	C₂×#123×C₆₇	1	48	yes	—
181	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
182	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
183	C₂²	C₂²×#41×C₆₇	1	4	yes	—
184	C₄	—	2	16	no	—
185	C₄	—	4	16	no	—
186	C₂	—	2	16	no	—
187	C₂	C₂×#381×C₆₇	1	24	yes	direct product factor
188	C₂	C₂³×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
189	C₂	C₂²×#117×C₆₇	1	2	yes	—
190	C₂	C₂²×#123×C₆₇	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₆₇	C₂³
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂⁶×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#381×C₆₇	C₂	direct (N × H)
4. C₂²×#95×C₆₇	C₂	semidirect
7. C₂⁴×C₄×C₆₇	C₄	semidirect
13. #381×C₆₇	C₂²	direct (N × H)
14. C₂×#95×C₆₇	C₂²	semidirect
26. C₂³×C₄×C₆₇	C₂×C₄	semidirect
32. #95×C₆₇	C₂³	semidirect
52. C₂²×C₄×C₆₇	C₂²×C₄	semidirect
80. C₂²×C₆₇	#381	direct (N × H)
90. C₂²×#381	C₆₇	direct (N × H)
92. C₂×C₆₇	C₂×#381	direct (N × H)
98. C₂×#381	C₂×C₆₇	direct (N × H)
99. C₂²×#95	C₂×C₆₇	semidirect
100. C₆₇	C₂²×#381	direct (N × H)
103. C₂⁴×C₄	C₄×C₆₇	semidirect
109. #381	C₂²×C₆₇	direct (N × H)
110. C₂×#95	C₂²×C₆₇	semidirect
122. C₂³×C₄	C₂×C₄×C₆₇	semidirect
128. #95	C₂³×C₆₇	semidirect
148. C₂²×C₄	C₂²×C₄×C₆₇	semidirect
176. C₂²	#381×C₆₇	direct (N × H)
187. C₂	C₂×#381×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₆₇	2
C₂³×C₂²⋊C₄×C₆₇	2
C₂×#381×C₆₇	2
C₂²×#95×C₆₇	2
C₂²×#381	67

Subgroups — C22×#381×C67 #30713256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#381×C₆₇ #30713256