Subgroups — C₂×#11725469 #13268643

Order 1792.

630 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#97	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6386	C₂	1	2	yes	maximal
3	#11725469	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₇×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
7	C₂×C₇×#97	—	2	4	no	—
8	C₂×C₇×#97	C₄	1	2	yes	—
9	C₂×C₇×#97	C₂²	1	2	yes	—
10	C₂×#6386	—	2	4	no	—
11	C₂×#6386	—	2	4	no	—
12	C₂×#6386	C₄	1	4	yes	complemented
13	C₂×#6386	C₂²	1	4	yes	—
14	C₂²×C₇⋊C₁₆	C₄	1	4	yes	complemented
15	C₂×#1375	—	7	1	no	maximal, Sylow
16	C₂³×C₄×C₇	—	2	1	no	—
17	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₇×C₈	—	2	2	no	—
19	C₂²×C₇×C₈	—	2	1	no	—
20	C₂²×C₇×C₈	—	2	1	no	—
21	C₂²×C₇×C₈	C₈	1	4	yes	—
22	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₇×C₈	D₄	1	2	yes	—
27	C₂²×C₇×C₈	D₄	1	4	yes	—
28	C₂²×C₇×C₈	Q₈	1	2	yes	—
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₇	—	2	2	no	—
31	C₂×C₄²×C₇	—	2	1	no	—
32	C₂×C₄²×C₇	C₈	1	4	yes	—
33	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
34	C₇×#97	—	2	4	no	—
35	C₇×#97	—	2	4	no	—
36	#6386	—	2	8	no	—
37	#6386	—	2	4	no	—
38	#6386	—	2	4	no	—
39	C₂×C₇⋊C₁₆	—	2	8	no	—
40	C₂×C₇⋊C₁₆	C₂×C₄	1	8	yes	complemented
41	C₂²×M₃₂	—	7	2	no	—
42	C₂²×#97	D₇	1	1	yes	characteristic
43	#1375	—	7	4	no	—
44	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₇	—	2	4	no	—
46	C₂²×C₄×C₇	—	2	2	no	—
47	C₂²×C₄×C₇	—	4	4	no	—
48	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₇×C₈	—	2	4	no	—
59	C₂×C₇×C₈	—	2	4	no	—
60	C₂×C₇×C₈	—	2	2	no	—
61	C₂×C₇×C₈	—	2	2	no	—
62	C₂×C₇×C₈	C₄²	1	2	yes	—
63	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
64	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
65	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
66	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
67	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
68	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
69	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
70	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
73	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₇	—	2	4	no	—
75	C₄²×C₇	—	2	2	no	—
76	C₄²×C₇	C₂×C₈	1	4	yes	—
77	C₇⋊C₁₆	—	2	8	no	—
78	C₂³×C₈	Dic₇	1	1	yes	characteristic
79	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
80	C₂²×C₄²	Dic₇	1	1	yes	characteristic
81	C₂×M₃₂	—	7	4	no	—
82	C₂×M₃₂	—	7	4	no	—
83	C₂×M₃₂	—	14	4	no	—
84	C₂×M₃₂	—	14	4	no	—
85	C₂²×C₁₆	—	7	4	no	—
86	C₂×#97	—	2	4	no	—
87	C₂×#97	Dic₇	1	2	yes	—
88	C₂×#97	C₂×D₇	1	2	yes	—
89	C₂³×C₇	—	2	4	no	—
90	C₂³×C₇	#97	1	1	yes	characteristic
91	C₂³×C₇	#97	1	1	yes	characteristic
92	C₂³×C₇	C₂²⋊C₈	1	2	yes	—
93	C₂³×C₇	C₄⋊C₈	1	2	yes	—
94	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₇	—	2	4	no	—
96	C₂×C₄×C₇	—	4	4	no	—
97	C₂×C₄×C₇	—	4	8	no	—
98	C₂×C₄×C₇	#95	1	2	yes	—
99	C₂×C₄×C₇	#95	1	2	yes	—
100	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
101	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
102	C₂×C₄×C₇	#97	1	1	yes	characteristic
103	C₂×C₄×C₇	#97	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
106	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
107	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
108	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
109	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
110	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
116	C₇×C₈	—	2	4	no	—
117	C₇×C₈	#95	1	4	yes	—
118	C₇×C₈	C₂×C₂²⋊C₄	1	2	yes	—
119	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
120	C₂³×C₄	—	2	1	no	—
121	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	2	1	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	C₇⋊C₈	1	4	yes	—
126	C₂²×C₈	Dic₁₄	1	2	yes	—
127	C₂²×C₈	C₄×D₇	1	2	yes	—
128	C₂²×C₈	C₄×D₇	1	2	yes	—
129	C₂²×C₈	D₂₈	1	2	yes	—
130	C₂²×C₈	C₂×Dic₇	1	2	yes	—
131	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
132	C₂²×C₈	C₂²⋊D₇	1	4	yes	—
133	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
134	C₂×C₄²	—	2	1	no	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₇⋊C₈	1	4	yes	—
137	C₂×C₄²	C₂×Dic₇	1	2	yes	—
138	M₃₂	—	14	8	no	—
139	M₃₂	—	14	4	no	—
140	M₃₂	—	14	4	no	—
141	C₂×C₁₆	—	7	8	no	—
142	C₂×C₁₆	—	14	8	no	—
143	#97	—	2	4	no	—
144	#97	—	2	4	no	—
145	C₂²×C₇	—	2	2	no	—
146	C₂²×C₇	—	2	2	no	—
147	C₂²×C₇	—	2	4	no	—
148	C₂²×C₇	—	2	4	no	—
149	C₂²×C₇	#336	1	2	yes	—
150	C₂²×C₇	#336	1	2	yes	—
151	C₂²×C₇	#347	1	4	yes	—
152	C₂²×C₇	C₂×#97	1	1	yes	characteristic
153	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
154	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
155	C₄×C₇	—	4	4	no	—
156	C₄×C₇	#336	1	2	yes	—
157	C₄×C₇	#336	1	2	yes	—
158	C₄×C₇	C₂×#95	1	1	yes	characteristic
159	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
160	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
162	C₂⁴	#1136	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
167	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
168	C₂²×C₄	#1136	1	1	yes	characteristic
169	C₂²×C₄	#1136	1	1	yes	characteristic
170	C₂²×C₄	#1136	1	1	yes	characteristic
171	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
172	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
174	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	C₂×C₇⋊C₈	1	4	yes	—
181	C₂×C₈	C₄×Dic₇	1	2	yes	—
182	C₂×C₈	Dic₇⋊C₄	1	4	yes	—
183	C₂×C₈	Dic₇⋊C₄	1	4	yes	—
184	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
185	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
186	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
187	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
188	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
189	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
190	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
191	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	2	no	—
194	C₄²	C₂×C₇⋊C₈	1	4	yes	—
195	C₁₆	—	14	8	no	—
196	C₂×C₇	—	2	1	no	—
197	C₂×C₇	—	2	2	no	—
198	C₂×C₇	—	2	1	no	—
199	C₂×C₇	#1375	1	4	yes	complemented
200	C₂×C₇	C₂×#336	1	1	yes	characteristic
201	C₂×C₇	C₂×#347	1	2	yes	—
202	C₂³	—	2	4	no	—
203	C₂³	#6377	1	1	yes	characteristic
204	C₂³	#6377	1	1	yes	characteristic
205	C₂³	C₄⋊(C₇⋊C₈)	1	2	yes	—
206	C₂³	C₂²⋊(C₇⋊C₈)	1	2	yes	—
207	C₂³	C₂×#1136	1	1	yes	characteristic
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	8	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	C₄×C₇⋊C₈	1	2	yes	—
212	C₂×C₄	C₄×C₇⋊C₈	1	2	yes	—
213	C₂×C₄	#6377	1	1	yes	characteristic
214	C₂×C₄	#6377	1	1	yes	characteristic
215	C₂×C₄	C₄⋊(C₇⋊C₈)	1	2	yes	—
216	C₂×C₄	C₄⋊(C₇⋊C₈)	1	2	yes	—
217	C₂×C₄	C₄⋊(C₇⋊C₈)	1	2	yes	—
218	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	2	yes	—
219	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	2	yes	—
220	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	2	yes	—
221	C₂×C₄	C₇⋊#95	1	2	yes	—
222	C₂×C₄	C₇⋊#95	1	2	yes	—
223	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
225	C₂×C₄	C₂×#1136	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
228	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
229	C₈	—	2	4	no	—
230	C₈	C₇⋊#95	1	4	yes	—
231	C₈	C₂×Dic₇⋊C₄	1	2	yes	—
232	C₈	C₂×(C₂×D₇)⋊C₄	1	2	yes	—
233	C₇	C₂×#1375	1	1	yes	characteristic, complemented, Sylow
234	C₂²	—	2	4	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	#89261	1	4	yes	—
239	C₂²	#89319	1	2	yes	—
240	C₂²	#89319	1	2	yes	—
241	C₂²	C₂×#6377	1	1	yes	characteristic
242	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
243	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
244	C₄	—	4	4	no	—
245	C₄	#89319	1	2	yes	—
246	C₄	#89319	1	2	yes	—
247	C₄	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
248	C₄	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
249	C₄	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
250	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	—	2	2	no	—
254	C₂	#11725469	1	4	yes	direct product factor
255	C₂	C₂×#89261	1	2	yes	—
256	C₂	C₂×#89319	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂²×C₇×#97	C₂
socle	C₂³×C₇	#97
prime radical	C₂⁴×C₇	M₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂²×#97	D₇
hypocenter	C₇	C₂×#1375
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #11725469	C₂	direct (N × H)
12. C₂×#6386	C₄	semidirect
14. C₂²×C₇⋊C₁₆	C₄	semidirect
40. C₂×C₇⋊C₁₆	C₂×C₄	semidirect
199. C₂×C₇	#1375	semidirect
233. C₇	C₂×#1375	semidirect
254. C₂	#11725469	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×#97	2
C₂²×#6386	2
#11725469	2
C₂×#1375	7

Subgroups — C2×#11725469 #13268643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11725469 #13268643