Subgroups — C₂²×#89956 #14217835

Order 1792.

2790 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₈×Dic₇	C₂	1	2	yes	maximal
3	C₂²×#6409	C₂	1	2	yes	maximal
4	C₂×#89956	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#6391	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6389	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₇×Q₈	C₂²	1	2	yes	—
8	C₂²×C₇×Q₁₆	C₄	1	16	yes	complemented
9	C₂²×C₇×Q₁₆	C₂²	1	12	yes	—
10	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×Q₈×Dic₇	—	2	12	no	—
12	C₂×Q₈×Dic₇	C₂²	1	24	yes	—
13	C₂²×C₄⋊Dic₇	—	2	2	no	—
14	C₂²×C₄⋊Dic₇	C₂²	1	2	yes	—
15	C₂²×C₄×Dic₇	—	2	2	no	—
16	C₂²×C₄×Dic₇	C₂²	1	1	yes	characteristic
17	C₂³×C₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×#6409	C₂²	1	24	yes	—
19	#89956	C₂²	1	64	yes	direct product factor
20	C₂×#6391	C₂²	1	12	yes	—
21	C₂×#6389	C₂²	1	12	yes	—
22	C₂²×#441	—	7	1	no	maximal, Sylow
23	C₂²×C₇×Q₈	—	2	6	no	—
24	C₂²×C₇×Q₈	—	2	8	no	—
25	C₂²×C₇×Q₈	C₂×C₄	1	16	yes	—
26	C₂²×C₇×Q₈	C₂³	1	12	yes	—
27	C₂³×C₄×C₇	—	2	2	no	—
28	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₂³×Dic₇	—	2	2	no	—
30	C₂³×Dic₇	D₄	1	2	yes	—
31	C₂×C₇×Q₁₆	C₂×C₄	1	96	yes	complemented
32	C₂×C₇×Q₁₆	C₂³	1	16	yes	—
33	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
34	C₂²×C₇×C₈	C₂³	1	6	yes	—
35	Q₈×Dic₇	—	2	48	no	—
36	Q₈×Dic₇	C₂³	1	32	yes	—
37	C₂×C₄⋊Dic₇	—	2	6	no	—
38	C₂×C₄⋊Dic₇	—	2	24	no	—
39	C₂×C₄⋊Dic₇	C₂³	1	12	yes	—
40	C₂×C₄×Dic₇	—	2	24	no	—
41	C₂×C₄×Dic₇	D₄	1	6	yes	—
42	C₂×C₄×Dic₇	C₂³	1	6	yes	—
43	C₂²×C₇⋊C₈	—	2	4	no	—
44	C₂²×C₇⋊C₈	C₂³	1	6	yes	—
45	#6409	C₂³	1	32	yes	—
46	#6391	C₂³	1	16	yes	—
47	#6389	C₂³	1	16	yes	—
48	C₂³×Q₁₆	D₇	1	1	yes	characteristic
49	C₂²×C₄×Q₈	—	7	2	no	—
50	C₂×#441	—	7	24	no	—
51	C₂²×#106	—	7	1	no	—
52	C₂²×#103	—	7	2	no	—
53	C₂²×#97	—	7	1	no	—
54	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₇×Q₈	—	2	48	no	—
56	C₂×C₇×Q₈	—	2	12	no	—
57	C₂×C₇×Q₈	—	2	24	no	—
58	C₂×C₇×Q₈	C₂²×C₄	1	48	yes	—
59	C₂×C₇×Q₈	C₂⁴	1	8	yes	—
60	C₂²×C₄×C₇	—	2	16	no	—
61	C₂²×C₄×C₇	—	2	12	no	—
62	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
64	C₂²×C₄×C₇	#41	1	4	yes	—
65	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
66	C₂²×Dic₇	—	2	8	no	—
67	C₂²×Dic₇	—	2	12	no	—
68	C₂²×Dic₇	—	4	8	no	—
69	C₂²×Dic₇	C₂×D₄	1	12	yes	—
70	C₇×Q₁₆	C₂²×C₄	1	64	yes	complemented
71	C₂×C₇×C₈	C₂²×C₄	1	24	yes	—
72	C₂×C₇×C₈	C₂⁴	1	4	yes	—
73	C₄⋊Dic₇	—	2	12	no	—
74	C₄⋊Dic₇	—	2	32	no	—
75	C₄⋊Dic₇	C₂⁴	1	8	yes	—
76	C₄×Dic₇	—	2	32	no	—
77	C₄×Dic₇	C₂×D₄	1	12	yes	—
78	C₄×Dic₇	C₂⁴	1	4	yes	—
79	C₂×C₇⋊C₈	—	2	12	no	—
80	C₂×C₇⋊C₈	C₂⁴	1	4	yes	—
81	C₂³×Q₈	C₂×D₇	1	2	yes	—
82	C₂²×Q₁₆	Dic₇	1	16	yes	complemented
83	C₂²×Q₁₆	C₂×D₇	1	12	yes	—
84	C₂³×C₈	—	7	1	no	—
85	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
86	C₂×C₄×Q₈	—	7	24	no	—
87	C₂×C₄×Q₈	—	14	12	no	—
88	C₂²×C₄⋊C₄	—	7	2	no	—
89	C₂²×C₄⋊C₄	—	14	2	no	—
90	C₂²×C₄²	—	7	1	no	—
91	C₂²×C₄²	—	14	2	no	—
92	#441	—	7	64	no	—
93	C₂×#106	—	7	12	no	—
94	C₂×#103	—	7	24	no	—
95	C₂×#97	—	7	12	no	—
96	C₂³×C₇	C₄×D₄	1	4	yes	—
97	C₂³×C₇	#137	1	8	yes	—
98	C₂³×C₇	C₂²×D₄	1	3	yes	—
99	C₇×Q₈	—	2	48	no	—
100	C₇×Q₈	—	2	8	no	—
101	C₇×Q₈	C₂³×C₄	1	16	yes	—
102	C₂×C₄×C₇	—	2	48	no	—
103	C₂×C₄×C₇	—	2	8	no	—
104	C₂×C₄×C₇	C₄×D₄	1	12	yes	—
105	C₂×C₄×C₇	C₂³×C₄	1	6	yes	—
106	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
107	C₂×C₄×C₇	C₂×#41	1	6	yes	—
108	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
109	C₂×Dic₇	—	2	24	no	—
110	C₂×Dic₇	—	2	8	no	—
111	C₂×Dic₇	—	4	24	no	—
112	C₂×Dic₇	C₂²×D₄	1	8	yes	—
113	C₇×C₈	C₂³×C₄	1	8	yes	—
114	C₇⋊C₈	—	2	4	no	—
115	C₂²×Q₈	—	2	6	no	—
116	C₂²×Q₈	—	2	8	no	—
117	C₂²×Q₈	C₂×Dic₇	1	16	yes	—
118	C₂²×Q₈	C₂²×D₇	1	12	yes	—
119	C₂³×C₄	—	2	2	no	—
120	C₂³×C₄	—	7	2	no	—
121	C₂³×C₄	—	14	2	no	—
122	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
123	C₂×Q₁₆	C₂×Dic₇	1	96	yes	complemented
124	C₂×Q₁₆	C₂²×D₇	1	16	yes	—
125	C₂²×C₈	—	7	6	no	—
126	C₂²×C₈	—	14	4	no	—
127	C₂²×C₈	C₂×Dic₇	1	8	yes	—
128	C₂²×C₈	C₂²×D₇	1	6	yes	—
129	C₄×Q₈	—	7	32	no	—
130	C₄×Q₈	—	14	48	no	—
131	C₂×C₄⋊C₄	—	7	12	no	—
132	C₂×C₄⋊C₄	—	14	24	no	—
133	C₂×C₄⋊C₄	—	14	6	no	—
134	C₂×C₄²	—	7	6	no	—
135	C₂×C₄²	—	7	6	no	—
136	C₂×C₄²	—	14	24	no	—
137	#106	—	7	16	no	—
138	#103	—	7	32	no	—
139	#97	—	7	16	no	—
140	C₂²×C₇	#441	1	16	yes	complemented
141	C₂²×C₇	C₂×C₄×D₄	1	6	yes	—
142	C₂²×C₇	C₂×#137	1	12	yes	—
143	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
144	C₄×C₇	—	2	16	no	—
145	C₄×C₇	C₂×C₄×D₄	1	6	yes	—
146	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
147	C₄×C₇	C₂²×#41	1	1	yes	characteristic
148	Dic₇	—	2	8	no	—
149	Dic₇	—	4	8	no	—
150	C₂⁴	D₄×D₇	1	1	yes	characteristic
151	C₂×Q₈	—	2	48	no	—
152	C₂×Q₈	—	2	12	no	—
153	C₂×Q₈	—	2	24	no	—
154	C₂×Q₈	C₂²×Dic₇	1	48	yes	—
155	C₂×Q₈	C₂³×D₇	1	8	yes	—
156	C₂²×C₄	—	2	16	no	—
157	C₂²×C₄	—	2	12	no	—
158	C₂²×C₄	—	7	12	no	—
159	C₂²×C₄	—	14	8	no	—
160	C₂²×C₄	—	14	12	no	—
161	C₂²×C₄	—	28	8	no	—
162	C₂²×C₄	D₄×D₇	1	3	yes	—
163	C₂²×C₄	#1159	1	4	yes	—
164	C₂²×C₄	C₂²×Dic₇	1	4	yes	—
165	C₂²×C₄	C₂³×D₇	1	3	yes	—
166	Q₁₆	C₂²×Dic₇	1	64	yes	complemented
167	C₂×C₈	—	7	4	no	—
168	C₂×C₈	—	14	12	no	—
169	C₂×C₈	C₂²×Dic₇	1	24	yes	—
170	C₂×C₈	C₂³×D₇	1	4	yes	—
171	C₄⋊C₄	—	7	8	no	—
172	C₄⋊C₄	—	14	32	no	—
173	C₄⋊C₄	—	14	12	no	—
174	C₄²	—	7	12	no	—
175	C₄²	—	7	4	no	—
176	C₄²	—	14	32	no	—
177	C₂×C₇	C₂×#441	1	12	yes	complemented
178	C₂×C₇	C₂²×C₄×D₄	1	1	yes	characteristic
179	C₂×C₇	C₂²×#137	1	2	yes	—
180	C₂³	#6481	1	8	yes	—
181	C₂³	D₄×Dic₇	1	4	yes	—
182	C₂³	C₂×D₄×D₇	1	3	yes	—
183	Q₈	—	2	48	no	—
184	Q₈	—	2	8	no	—
185	Q₈	C₂³×Dic₇	1	16	yes	—
186	C₂×C₄	—	2	48	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	—	7	8	no	—
189	C₂×C₄	—	14	24	no	—
190	C₂×C₄	—	14	8	no	—
191	C₂×C₄	—	28	24	no	—
192	C₂×C₄	D₄×Dic₇	1	12	yes	—
193	C₂×C₄	C₂×D₄×D₇	1	3	yes	—
194	C₂×C₄	C₂×#1159	1	6	yes	—
195	C₂×C₄	C₂³×Dic₇	1	6	yes	—
196	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
197	C₈	—	14	4	no	—
198	C₈	C₂³×Dic₇	1	8	yes	—
199	C₇	C₂²×#441	1	1	yes	characteristic, complemented, Sylow
200	C₂²	#89956	1	16	yes	direct product factor
201	C₂²	C₂×#6481	1	12	yes	—
202	C₂²	C₂×D₄×Dic₇	1	6	yes	—
203	C₂²	C₂²×D₄×D₇	1	1	yes	characteristic
204	C₄	—	2	16	no	—
205	C₄	—	14	8	no	—
206	C₄	—	28	8	no	—
207	C₄	C₂×D₄×Dic₇	1	6	yes	—
208	C₄	C₂²×#1159	1	1	yes	characteristic
209	C₄	C₂⁴×Dic₇	1	1	yes	characteristic
210	C₂	C₂×#89956	1	12	yes	direct product factor
211	C₂	C₂²×#6481	1	2	yes	—
212	C₂	C₂²×D₄×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₇
commutator	C₄×C₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇
Fitting	C₂³×C₇×Q₁₆	C₂
socle, prime radical	C₂⁴×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₂³×Q₁₆	D₇
hypocenter	C₇	C₂²×#441
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
4. C₂×#89956	C₂	direct (N × H)
8. C₂²×C₇×Q₁₆	C₄	semidirect
19. #89956	C₂²	direct (N × H)
31. C₂×C₇×Q₁₆	C₂×C₄	semidirect
70. C₇×Q₁₆	C₂²×C₄	semidirect
82. C₂²×Q₁₆	Dic₇	semidirect
123. C₂×Q₁₆	C₂×Dic₇	semidirect
140. C₂²×C₇	#441	semidirect
166. Q₁₆	C₂²×Dic₇	semidirect
177. C₂×C₇	C₂×#441	semidirect
199. C₇	C₂²×#441	semidirect
200. C₂²	#89956	direct (N × H)
210. C₂	C₂×#89956	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×Q₁₆	2
C₂²×Q₈×Dic₇	2
C₂²×#6409	2
C₂×#89956	2
C₂²×#6391	2
C₂²×#6389	2
C₂²×#441	7

Subgroups — C22×#89956 #14217835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#89956 #14217835