Subgroups — C₂³×#6396 #14217484

Order 1792.

2846 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂³×#1136	C₂	1	1	yes	maximal, characteristic
3	C₂⁴×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×#6396	C₂	1	28	yes	maximal, direct product factor
5	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂²×C₇×M₁₆	—	2	7	no	—
7	C₂²×C₇×M₁₆	C₂²	1	14	yes	—
8	C₂³×C₇×C₈	—	2	1	no	—
9	C₂²×#1136	—	2	7	no	—
10	C₂²×#1136	C₂²	1	14	yes	—
11	C₂³×C₇⋊C₈	—	2	1	no	—
12	C₂³×C₇⋊C₈	C₄	1	14	yes	—
13	C₂³×C₇⋊C₈	C₄	1	2	yes	—
14	C₂³×C₇⋊C₈	C₂²	1	14	yes	—
15	C₂×#6396	C₂²	1	112	yes	direct product factor
16	C₂³×#108	—	7	1	no	maximal, Sylow
17	C₂⁵×C₇	Q₈	1	1	yes	characteristic
18	C₂³×C₄×C₇	C₂×C₄	1	7	yes	—
19	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₇	D₄	1	7	yes	—
21	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₇	Q₈	1	7	yes	—
23	C₂³×C₄×C₇	C₂³	1	7	yes	—
24	C₂×C₇×M₁₆	—	2	42	no	—
25	C₂×C₇×M₁₆	C₂³	1	28	yes	—
26	C₂²×C₇×C₈	—	2	14	no	—
27	C₂×#1136	—	2	42	no	—
28	C₂×#1136	C₂³	1	28	yes	—
29	C₂²×C₇⋊C₈	—	2	14	no	—
30	C₂²×C₇⋊C₈	C₂×C₄	1	84	yes	—
31	C₂²×C₇⋊C₈	C₂×C₄	1	28	yes	—
32	C₂²×C₇⋊C₈	C₂³	1	28	yes	—
33	#6396	C₂³	1	64	yes	direct product factor
34	C₂³×M₁₆	—	7	1	no	—
35	C₂³×M₁₆	D₇	1	1	yes	characteristic
36	C₂⁴×C₈	—	7	1	no	—
37	C₂²×#108	—	7	28	no	—
38	C₂⁴×C₇	—	2	8	no	—
39	C₂⁴×C₇	C₄⋊C₄	1	7	yes	—
40	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂⁴×C₇	C₂×Q₈	1	7	yes	—
42	C₂²×C₄×C₇	C₄⋊C₄	1	42	yes	—
43	C₂²×C₄×C₇	C₄⋊C₄	1	7	yes	—
44	C₂²×C₄×C₇	C₄⋊C₄	1	7	yes	—
45	C₂²×C₄×C₇	C₂²×C₄	1	21	yes	—
46	C₂²×C₄×C₇	C₂²×C₄	1	7	yes	—
47	C₂²×C₄×C₇	C₂×D₄	1	7	yes	—
48	C₂²×C₄×C₇	C₂×D₄	1	21	yes	—
49	C₂²×C₄×C₇	C₂×Q₈	1	21	yes	—
50	C₂²×C₄×C₇	C₂⁴	1	7	yes	—
51	C₇×M₁₆	—	2	28	no	—
52	C₇×M₁₆	C₂⁴	1	8	yes	—
53	C₂×C₇×C₈	—	2	28	no	—
54	#1136	—	2	28	no	—
55	#1136	C₂⁴	1	8	yes	—
56	C₂×C₇⋊C₈	—	2	28	no	—
57	C₂×C₇⋊C₈	C₂²×C₄	1	56	yes	—
58	C₂×C₇⋊C₈	C₂²×C₄	1	56	yes	—
59	C₂×C₇⋊C₈	C₂⁴	1	8	yes	—
60	C₂⁴×C₄	C₂×D₇	1	1	yes	characteristic
61	C₂²×M₁₆	—	2	7	no	—
62	C₂²×M₁₆	—	7	14	no	—
63	C₂²×M₁₆	—	14	7	no	—
64	C₂²×M₁₆	C₂×D₇	1	14	yes	—
65	C₂³×C₈	—	2	1	no	—
66	C₂³×C₈	—	7	2	no	—
67	C₂³×C₈	—	7	14	no	—
68	C₂³×C₈	—	7	14	no	—
69	C₂³×C₈	—	14	1	no	—
70	C₂×#108	—	7	112	no	—
71	C₂³×C₇	—	2	28	no	—
72	C₂³×C₇	—	2	28	no	—
73	C₂³×C₇	#108	1	8	yes	complemented
74	C₂³×C₇	C₂×C₄⋊C₄	1	21	yes	—
75	C₂³×C₇	C₂×C₄⋊C₄	1	7	yes	—
76	C₂³×C₇	C₂²×Q₈	1	7	yes	—
77	C₂×C₄×C₇	C₂×C₄⋊C₄	1	42	yes	—
78	C₂×C₄×C₇	C₂×C₄⋊C₄	1	21	yes	—
79	C₂×C₄×C₇	C₂×C₄⋊C₄	1	21	yes	—
80	C₂×C₄×C₇	C₂³×C₄	1	7	yes	—
81	C₂×C₄×C₇	C₂³×C₄	1	7	yes	—
82	C₂×C₄×C₇	C₂²×D₄	1	7	yes	—
83	C₂×C₄×C₇	C₂²×D₄	1	7	yes	—
84	C₂×C₄×C₇	C₂²×Q₈	1	7	yes	—
85	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
86	C₇×C₈	—	2	8	no	—
87	C₇⋊C₈	—	2	8	no	—
88	C₇⋊C₈	C₂³×C₄	1	16	yes	—
89	C₂⁵	Dic₁₄	1	1	yes	characteristic
90	C₂³×C₄	Dic₁₄	1	7	yes	—
91	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
92	C₂³×C₄	C₄×D₇	1	7	yes	—
93	C₂³×C₄	C₂²⋊D₇	1	7	yes	—
94	C₂³×C₄	C₂²⋊D₇	1	1	yes	characteristic
95	C₂³×C₄	C₂²×D₇	1	7	yes	—
96	C₂×M₁₆	—	2	42	no	—
97	C₂×M₁₆	—	7	28	no	—
98	C₂×M₁₆	—	14	42	no	—
99	C₂×M₁₆	C₂²×D₇	1	28	yes	—
100	C₂²×C₈	—	2	14	no	—
101	C₂²×C₈	—	7	84	no	—
102	C₂²×C₈	—	7	28	no	—
103	C₂²×C₈	—	7	28	no	—
104	C₂²×C₈	—	14	14	no	—
105	#108	—	7	64	no	—
106	C₂²×C₇	—	2	42	no	—
107	C₂²×C₇	—	2	14	no	—
108	C₂²×C₇	C₂×#108	1	28	yes	complemented
109	C₂²×C₇	C₂²×C₄⋊C₄	1	7	yes	—
110	C₂²×C₇	C₂²×C₄⋊C₄	1	7	yes	—
111	C₂²×C₇	C₂³×Q₈	1	1	yes	characteristic
112	C₄×C₇	C₂²×C₄⋊C₄	1	7	yes	—
113	C₄×C₇	C₂²×C₄⋊C₄	1	7	yes	—
114	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
115	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
116	C₂⁴	—	2	8	no	—
117	C₂⁴	Dic₇⋊C₄	1	7	yes	—
118	C₂⁴	Dic₇⋊C₄	1	1	yes	characteristic
119	C₂⁴	C₂×Dic₁₄	1	7	yes	—
120	C₂²×C₄	Dic₇⋊C₄	1	42	yes	—
121	C₂²×C₄	Dic₇⋊C₄	1	7	yes	—
122	C₂²×C₄	Dic₇⋊C₄	1	7	yes	—
123	C₂²×C₄	C₂×Dic₁₄	1	21	yes	—
124	C₂²×C₄	C₂×C₄×D₇	1	21	yes	—
125	C₂²×C₄	C₂×C₄×D₇	1	7	yes	—
126	C₂²×C₄	C₂×C₂²⋊D₇	1	7	yes	—
127	C₂²×C₄	C₂×C₂²⋊D₇	1	21	yes	—
128	C₂²×C₄	C₂³×D₇	1	7	yes	—
129	M₁₆	—	2	28	no	—
130	M₁₆	—	7	8	no	—
131	M₁₆	—	14	28	no	—
132	M₁₆	C₂³×D₇	1	8	yes	—
133	C₂×C₈	—	2	28	no	—
134	C₂×C₈	—	7	56	no	—
135	C₂×C₈	—	7	56	no	—
136	C₂×C₈	—	7	8	no	—
137	C₂×C₈	—	14	28	no	—
138	C₂×C₇	—	2	7	no	—
139	C₂×C₇	—	2	1	no	—
140	C₂×C₇	C₂²×#108	1	14	yes	complemented
141	C₂×C₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
142	C₂³	—	2	28	no	—
143	C₂³	—	2	28	no	—
144	C₂³	#6396	1	8	yes	direct product factor
145	C₂³	C₂×Dic₇⋊C₄	1	21	yes	—
146	C₂³	C₂×Dic₇⋊C₄	1	7	yes	—
147	C₂³	C₂²×Dic₁₄	1	7	yes	—
148	C₂×C₄	C₂×Dic₇⋊C₄	1	42	yes	—
149	C₂×C₄	C₂×Dic₇⋊C₄	1	21	yes	—
150	C₂×C₄	C₂×Dic₇⋊C₄	1	21	yes	—
151	C₂×C₄	C₂²×Dic₁₄	1	7	yes	—
152	C₂×C₄	C₂²×C₄×D₇	1	7	yes	—
153	C₂×C₄	C₂²×C₄×D₇	1	7	yes	—
154	C₂×C₄	C₂²×C₂²⋊D₇	1	7	yes	—
155	C₂×C₄	C₂²×C₂²⋊D₇	1	7	yes	—
156	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
157	C₈	—	2	8	no	—
158	C₈	—	7	16	no	—
159	C₈	—	14	8	no	—
160	C₇	C₂³×#108	1	1	yes	characteristic, complemented, Sylow
161	C₂²	—	2	42	no	—
162	C₂²	—	2	14	no	—
163	C₂²	C₂×#6396	1	28	yes	direct product factor
164	C₂²	C₂²×Dic₇⋊C₄	1	7	yes	—
165	C₂²	C₂²×Dic₇⋊C₄	1	7	yes	—
166	C₂²	C₂³×Dic₁₄	1	1	yes	characteristic
167	C₄	C₂²×Dic₇⋊C₄	1	7	yes	—
168	C₄	C₂²×Dic₇⋊C₄	1	7	yes	—
169	C₄	C₂³×C₄×D₇	1	1	yes	characteristic
170	C₄	C₂³×C₂²⋊D₇	1	1	yes	characteristic
171	C₂	—	2	7	no	—
172	C₂	—	2	1	no	—
173	C₂	C₂²×#6396	1	14	yes	direct product factor
174	C₂	C₂³×Dic₇⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₇
commutator	C₄×C₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇
Fitting	C₂³×C₇×M₁₆	C₂
socle	C₂⁴×C₇	C₄⋊C₄
prime radical	C₂⁵×C₇	Q₈
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₂³×M₁₆	D₇
hypocenter	C₇	C₂³×#108
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₂²×#6396	C₂	direct (N × H)
15. C₂×#6396	C₂²	direct (N × H)
33. #6396	C₂³	direct (N × H)
73. C₂³×C₇	#108	semidirect
108. C₂²×C₇	C₂×#108	semidirect
140. C₂×C₇	C₂²×#108	semidirect
144. C₂³	#6396	direct (N × H)
160. C₇	C₂³×#108	semidirect
163. C₂²	C₂×#6396	direct (N × H)
173. C₂	C₂²×#6396	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×M₁₆	2
C₂³×#1136	2
C₂⁴×C₇⋊C₈	2
C₂²×#6396	2
C₂³×#108	7

Subgroups — C23×#6396 #14217484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6396 #14217484