Subgroups — C₂³×#14523967 #31719343

Order 17152.

5502 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14523967	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₆₇	—	2	1	no	—
6	C₂³×C₆₇⋊C₈	—	2	8	no	—
7	C₂³×C₆₇⋊C₈	C₂²	1	14	yes	—
8	C₂²×C₄⋊Dic₆₇	—	2	7	no	—
9	C₂²×C₄⋊Dic₆₇	C₂²	1	14	yes	—
10	C₂×#14523967	C₂²	1	112	yes	direct product factor
11	C₂³×Q₈×C₆₇	—	2	8	no	—
12	C₂³×Q₈×C₆₇	—	2	7	no	—
13	C₂³×Q₈×C₆₇	C₄	1	16	yes	complemented
14	C₂³×Q₈×C₆₇	C₂²	1	14	yes	—
15	C₂⁴×C₄×C₆₇	—	2	1	no	—
16	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×Dic₆₇	—	2	14	no	—
18	C₂³×Dic₆₇	—	4	8	no	—
19	C₂²×C₆₇⋊C₈	—	2	56	no	—
20	C₂²×C₆₇⋊C₈	C₂³	1	28	yes	—
21	C₂×C₄⋊Dic₆₇	—	2	42	no	—
22	C₂×C₄⋊Dic₆₇	C₂³	1	28	yes	—
23	#14523967	C₂³	1	64	yes	direct product factor
24	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
25	C₂²×Q₈×C₆₇	—	2	112	no	—
26	C₂²×Q₈×C₆₇	—	2	56	no	—
27	C₂²×Q₈×C₆₇	—	2	42	no	—
28	C₂²×Q₈×C₆₇	C₂×C₄	1	112	yes	complemented
29	C₂²×Q₈×C₆₇	C₂³	1	28	yes	—
30	C₂³×C₄×C₆₇	—	2	16	no	—
31	C₂³×C₄×C₆₇	—	2	14	no	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₆₇	D₄	1	7	yes	—
34	C₂³×C₄×C₆₇	D₄	1	8	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
36	C₂²×Dic₆₇	—	2	28	no	—
37	C₂²×Dic₆₇	—	4	56	no	—
38	C₂×C₆₇⋊C₈	—	2	56	no	—
39	C₂×C₆₇⋊C₈	C₂⁴	1	8	yes	—
40	C₄⋊Dic₆₇	—	2	28	no	—
41	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
42	C₂⁴×C₆₇	C₂²⋊C₄	1	8	yes	—
43	C₂⁴×C₆₇	SD₁₆	1	8	yes	—
44	C₂⁴×C₆₇	Q₁₆	1	8	yes	—
45	C₂⁴×C₆₇	C₂×D₄	1	7	yes	—
46	C₂×Q₈×C₆₇	—	2	336	no	—
47	C₂×Q₈×C₆₇	—	2	56	no	—
48	C₂×Q₈×C₆₇	—	2	28	no	—
49	C₂×Q₈×C₆₇	C₂²×C₄	1	112	yes	complemented
50	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
51	C₂²×C₄×C₆₇	—	2	112	no	—
52	C₂²×C₄×C₆₇	—	2	28	no	—
53	C₂²×C₄×C₆₇	C₂²⋊C₄	1	56	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	28	yes	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	21	yes	—
56	C₂²×C₄×C₆₇	C₂×D₄	1	28	yes	—
57	C₂²×C₄×C₆₇	C₂⁴	1	7	yes	—
58	C₂³×C₆₇	#103	1	64	yes	complemented
59	C₂³×C₆₇	C₂×C₂²⋊C₄	1	28	yes	—
60	C₂³×C₆₇	C₂×SD₁₆	1	28	yes	—
61	C₂³×C₆₇	C₂×Q₁₆	1	28	yes	—
62	C₂³×C₆₇	C₂²×D₄	1	7	yes	—
63	Q₈×C₆₇	—	2	112	no	—
64	Q₈×C₆₇	—	2	8	no	—
65	Q₈×C₆₇	C₂³×C₄	1	16	yes	complemented
66	C₂×C₄×C₆₇	—	2	112	no	—
67	C₂×C₄×C₆₇	—	2	8	no	—
68	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	84	yes	—
69	C₂×C₄×C₆₇	C₂³×C₄	1	14	yes	—
70	C₂×C₄×C₆₇	C₂²×D₄	1	7	yes	—
71	C₂×C₄×C₆₇	C₂²×D₄	1	14	yes	—
72	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
73	C₂×Dic₆₇	—	2	8	no	—
74	C₂×Dic₆₇	—	4	56	no	—
75	C₆₇⋊C₈	—	2	8	no	—
76	C₂²×C₆₇	C₂×#103	1	112	yes	complemented
77	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	14	yes	—
78	C₂²×C₆₇	C₂²×SD₁₆	1	14	yes	—
79	C₂²×C₆₇	C₂²×Q₁₆	1	14	yes	—
80	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
81	C₄×C₆₇	—	2	16	no	—
82	C₄×C₆₇	C₂²×C₂²⋊C₄	1	14	yes	—
83	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
84	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
85	Dic₆₇	—	4	8	no	—
86	C₂³×#103	—	67	1	no	maximal, Sylow
87	C₂×C₆₇	C₂²×#103	1	28	yes	complemented
88	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₆₇	C₂³×SD₁₆	1	1	yes	characteristic
90	C₂×C₆₇	C₂³×Q₁₆	1	1	yes	characteristic
91	C₂⁴×Q₈	D₆₇	1	1	yes	characteristic
92	C₂⁴×C₈	—	67	1	no	—
93	C₂³×C₄⋊C₄	—	67	1	no	—
94	C₂²×#103	—	67	28	no	—
95	C₆₇	C₂³×#103	1	1	yes	characteristic, complemented, Sylow
96	C₂³×Q₈	—	2	8	no	—
97	C₂³×Q₈	—	2	7	no	—
98	C₂³×Q₈	Dic₆₇	1	16	yes	complemented
99	C₂³×Q₈	C₂×D₆₇	1	14	yes	—
100	C₂⁴×C₄	—	2	1	no	—
101	C₂⁴×C₄	—	134	1	no	—
102	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
103	C₂³×C₈	—	67	14	no	—
104	C₂³×C₈	—	134	8	no	—
105	C₂²×C₄⋊C₄	—	67	14	no	—
106	C₂²×C₄⋊C₄	—	134	7	no	—
107	C₂×#103	—	67	112	no	—
108	C₂⁵	C₂²⋊D₆₇	1	1	yes	characteristic
109	C₂²×Q₈	—	2	112	no	—
110	C₂²×Q₈	—	2	56	no	—
111	C₂²×Q₈	—	2	42	no	—
112	C₂²×Q₈	C₂×Dic₆₇	1	112	yes	complemented
113	C₂²×Q₈	C₂²×D₆₇	1	28	yes	—
114	C₂³×C₄	—	2	16	no	—
115	C₂³×C₄	—	2	14	no	—
116	C₂³×C₄	—	268	8	no	—
117	C₂³×C₄	—	134	14	no	—
118	C₂³×C₄	C₂×Dic₆₇	1	8	yes	—
119	C₂³×C₄	C₂²⋊D₆₇	1	7	yes	—
120	C₂³×C₄	C₂²⋊D₆₇	1	8	yes	—
121	C₂³×C₄	C₂²×D₆₇	1	7	yes	—
122	C₂²×C₈	—	67	28	no	—
123	C₂²×C₈	—	134	56	no	—
124	C₂×C₄⋊C₄	—	67	28	no	—
125	C₂×C₄⋊C₄	—	134	42	no	—
126	#103	—	67	64	no	—
127	C₂⁴	C₂²⋊Dic₆₇	1	8	yes	—
128	C₂⁴	#11761615	1	8	yes	—
129	C₂⁴	#11761618	1	8	yes	—
130	C₂⁴	C₂×C₂²⋊D₆₇	1	7	yes	—
131	C₂×Q₈	—	2	336	no	—
132	C₂×Q₈	—	2	56	no	—
133	C₂×Q₈	—	2	28	no	—
134	C₂×Q₈	C₂²×Dic₆₇	1	112	yes	complemented
135	C₂×Q₈	C₂³×D₆₇	1	8	yes	—
136	C₂²×C₄	—	2	112	no	—
137	C₂²×C₄	—	2	28	no	—
138	C₂²×C₄	—	268	56	no	—
139	C₂²×C₄	—	134	28	no	—
140	C₂²×C₄	C₂²⋊Dic₆₇	1	56	yes	—
141	C₂²×C₄	C₂²×Dic₆₇	1	28	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₆₇	1	21	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₆₇	1	28	yes	—
144	C₂²×C₄	C₂³×D₆₇	1	7	yes	—
145	C₂×C₈	—	67	8	no	—
146	C₂×C₈	—	134	56	no	—
147	C₄⋊C₄	—	67	8	no	—
148	C₄⋊C₄	—	134	28	no	—
149	C₂³	#14523967	1	64	yes	direct product factor
150	C₂³	C₂×C₂²⋊Dic₆₇	1	28	yes	—
151	C₂³	C₂×#11761615	1	28	yes	—
152	C₂³	C₂×#11761618	1	28	yes	—
153	C₂³	C₂²×C₂²⋊D₆₇	1	7	yes	—
154	Q₈	—	2	112	no	—
155	Q₈	—	2	8	no	—
156	Q₈	C₂³×Dic₆₇	1	16	yes	complemented
157	C₂×C₄	—	2	112	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	268	56	no	—
160	C₂×C₄	—	134	8	no	—
161	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	84	yes	—
162	C₂×C₄	C₂³×Dic₆₇	1	14	yes	—
163	C₂×C₄	C₂²×C₂²⋊D₆₇	1	7	yes	—
164	C₂×C₄	C₂²×C₂²⋊D₆₇	1	14	yes	—
165	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
166	C₈	—	134	8	no	—
167	C₂²	C₂×#14523967	1	112	yes	direct product factor
168	C₂²	C₂²×C₂²⋊Dic₆₇	1	14	yes	—
169	C₂²	C₂²×#11761615	1	14	yes	—
170	C₂²	C₂²×#11761618	1	14	yes	—
171	C₂²	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
172	C₄	—	2	16	no	—
173	C₄	—	268	8	no	—
174	C₄	C₂²×C₂²⋊Dic₆₇	1	14	yes	—
175	C₄	C₂⁴×Dic₆₇	1	1	yes	characteristic
176	C₄	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
177	C₂	C₂²×#14523967	1	28	yes	direct product factor
178	C₂	C₂³×C₂²⋊Dic₆₇	1	1	yes	characteristic
179	C₂	C₂³×#11761615	1	1	yes	characteristic
180	C₂	C₂³×#11761618	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂⁴×Q₈×C₆₇	C₂
socle, prime radical	C₂⁵×C₆₇	D₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂⁴×Q₈	D₆₇
hypocenter	C₆₇	C₂³×#103
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×Q₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#14523967	C₂	direct (N × H)
10. C₂×#14523967	C₂²	direct (N × H)
13. C₂³×Q₈×C₆₇	C₄	semidirect
23. #14523967	C₂³	direct (N × H)
28. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
49. C₂×Q₈×C₆₇	C₂²×C₄	semidirect
58. C₂³×C₆₇	#103	semidirect
65. Q₈×C₆₇	C₂³×C₄	semidirect
76. C₂²×C₆₇	C₂×#103	semidirect
87. C₂×C₆₇	C₂²×#103	semidirect
95. C₆₇	C₂³×#103	semidirect
98. C₂³×Q₈	Dic₆₇	semidirect
112. C₂²×Q₈	C₂×Dic₆₇	semidirect
134. C₂×Q₈	C₂²×Dic₆₇	semidirect
149. C₂³	#14523967	direct (N × H)
156. Q₈	C₂³×Dic₆₇	semidirect
167. C₂²	C₂×#14523967	direct (N × H)
177. C₂	C₂²×#14523967	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₆₇⋊C₈	2
C₂³×C₄⋊Dic₆₇	2
C₂²×#14523967	2
C₂⁴×Q₈×C₆₇	2
C₂³×#103	67

Subgroups — C23×#14523967 #31719343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14523967 #31719343