Subgroups — C₂×#11725470 #13268655

Order 1792.

630 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#11725470	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	2	no	—
8	C₂×C₇×#97	—	2	2	no	—
9	C₂×C₇×#97	C₄	1	2	yes	—
10	C₂×C₇×#97	C₂²	1	2	yes	—
11	C₂×#6386	—	2	4	no	—
12	C₂×#6386	—	2	2	no	—
13	C₂×#6386	C₄	1	8	yes	complemented
14	C₂×#6386	C₂²	1	4	yes	—
15	C₂²×C₇⋊C₁₆	—	2	2	no	—
16	C₂×#1377	—	7	1	no	maximal, Sylow
17	C₂³×C₄×C₇	C₈	1	2	yes	—
18	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	—	2	1	no	—
20	C₂²×C₇×C₈	—	2	1	no	—
21	C₂²×C₇×C₈	—	2	2	no	—
22	C₂²×C₇×C₈	C₈	1	4	yes	—
23	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₇×C₈	D₄	1	2	yes	—
27	C₂²×C₇×C₈	D₄	1	2	yes	—
28	C₂²×C₇×C₈	D₄	1	2	yes	—
29	C₂²×C₇×C₈	Q₈	1	2	yes	—
30	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇	—	2	1	no	—
32	C₂×C₄²×C₇	—	2	1	no	—
33	C₂×C₄²×C₇	—	2	1	no	—
34	C₂×C₄²×C₇	—	2	1	no	—
35	C₂×C₄²×C₇	C₈	1	2	yes	—
36	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
37	C₇×#97	—	2	4	no	—
38	C₇×#97	—	2	4	no	—
39	#6386	—	2	8	no	—
40	#6386	—	2	4	no	—
41	#6386	C₂×C₄	1	8	yes	complemented
42	C₂×C₇⋊C₁₆	—	2	8	no	—
43	C₂×C₇⋊C₁₆	—	2	4	no	—
44	C₂²×M₃₂	—	7	2	no	—
45	C₂²×#97	D₇	1	1	yes	characteristic
46	#1377	—	7	4	no	—
47	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₇	—	2	2	no	—
49	C₂²×C₄×C₇	—	2	2	no	—
50	C₂²×C₄×C₇	—	4	4	no	—
51	C₂²×C₄×C₇	C₄²	1	2	yes	—
52	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
56	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₇	M₁₆	1	2	yes	—
59	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₇×C₈	—	2	4	no	—
63	C₂×C₇×C₈	—	2	4	no	—
64	C₂×C₇×C₈	—	2	2	no	—
65	C₂×C₇×C₈	—	2	2	no	—
66	C₂×C₇×C₈	C₄²	1	2	yes	—
67	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
68	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
69	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
70	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
71	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
72	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
73	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
74	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
75	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₇	—	2	2	no	—
79	C₄²×C₇	—	2	2	no	—
80	C₄²×C₇	—	2	2	no	—
81	C₄²×C₇	—	2	2	no	—
82	C₇⋊C₁₆	—	2	8	no	—
83	C₂³×C₈	Dic₇	1	1	yes	characteristic
84	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
85	C₂²×C₄²	Dic₇	1	1	yes	characteristic
86	C₂×M₃₂	—	7	8	no	—
87	C₂×M₃₂	—	7	4	no	—
88	C₂×M₃₂	—	14	4	no	—
89	C₂×M₃₂	—	14	2	no	—
90	C₂²×C₁₆	—	14	2	no	—
91	C₂×#97	—	2	2	no	—
92	C₂×#97	—	2	2	no	—
93	C₂×#97	Dic₇	1	2	yes	—
94	C₂×#97	C₂×D₇	1	2	yes	—
95	C₂³×C₇	—	2	4	no	—
96	C₂³×C₇	C₄×C₈	1	2	yes	—
97	C₂³×C₇	C₂²⋊C₈	1	1	yes	characteristic
98	C₂³×C₇	C₂²⋊C₈	1	1	yes	characteristic
99	C₂³×C₇	C₄⋊C₈	1	2	yes	—
100	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₇	—	2	4	no	—
102	C₂×C₄×C₇	—	4	4	no	—
103	C₂×C₄×C₇	—	4	4	no	—
104	C₂×C₄×C₇	—	4	4	no	—
105	C₂×C₄×C₇	#95	1	2	yes	—
106	C₂×C₄×C₇	#95	1	2	yes	—
107	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
108	C₂×C₄×C₇	#97	1	2	yes	—
109	C₂×C₄×C₇	#97	1	2	yes	—
110	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
113	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
114	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
115	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
116	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
117	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
123	C₇×C₈	—	2	4	no	—
124	C₇×C₈	#95	1	4	yes	—
125	C₇×C₈	C₂×C₂²⋊C₄	1	2	yes	—
126	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
127	C₂³×C₄	C₇⋊C₈	1	2	yes	—
128	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	2	1	no	—
131	C₂²×C₈	—	2	1	no	—
132	C₂²×C₈	C₇⋊C₈	1	4	yes	—
133	C₂²×C₈	Dic₁₄	1	2	yes	—
134	C₂²×C₈	C₄×D₇	1	4	yes	—
135	C₂²×C₈	D₂₈	1	2	yes	—
136	C₂²×C₈	C₂×Dic₇	1	2	yes	—
137	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
138	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
139	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
140	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	—	2	1	no	—
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	C₇⋊C₈	1	2	yes	—
146	C₂×C₄²	C₂×Dic₇	1	2	yes	—
147	M₃₂	—	7	8	no	—
148	M₃₂	—	14	8	no	—
149	M₃₂	—	14	4	no	—
150	C₂×C₁₆	—	14	8	no	—
151	C₂×C₁₆	—	14	4	no	—
152	#97	—	2	4	no	—
153	#97	—	2	4	no	—
154	C₂²×C₇	—	2	2	no	—
155	C₂²×C₇	—	2	2	no	—
156	C₂²×C₇	—	2	4	no	—
157	C₂²×C₇	—	2	4	no	—
158	C₂²×C₇	#336	1	2	yes	—
159	C₂²×C₇	#336	1	2	yes	—
160	C₂²×C₇	#349	1	4	yes	—
161	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
162	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
164	C₄×C₇	—	4	4	no	—
165	C₄×C₇	#336	1	2	yes	—
166	C₄×C₇	#336	1	2	yes	—
167	C₄×C₇	C₂×#95	1	1	yes	characteristic
168	C₄×C₇	C₂×#97	1	1	yes	characteristic
169	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
171	C₂⁴	C₂×C₇⋊C₈	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	4	no	—
175	C₂²×C₄	C₂×C₇⋊C₈	1	4	yes	—
176	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
177	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
178	C₂²×C₄	#1136	1	2	yes	—
179	C₂²×C₄	#1136	1	1	yes	characteristic
180	C₂²×C₄	#1136	1	1	yes	characteristic
181	C₂²×C₄	C₄×Dic₇	1	2	yes	—
182	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
183	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
185	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	C₂×C₇⋊C₈	1	4	yes	—
191	C₂×C₈	C₄×Dic₇	1	2	yes	—
192	C₂×C₈	Dic₇⋊C₄	1	4	yes	—
193	C₂×C₈	Dic₇⋊C₄	1	4	yes	—
194	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
195	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
196	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
197	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
198	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
200	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
201	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₁₆	—	14	8	no	—
207	C₂×C₇	—	2	2	no	—
208	C₂×C₇	—	2	1	no	—
209	C₂×C₇	—	2	1	no	—
210	C₂×C₇	#1377	1	4	yes	complemented
211	C₂×C₇	C₂×#336	1	1	yes	characteristic
212	C₂×C₇	C₂×#349	1	2	yes	—
213	C₂³	—	2	4	no	—
214	C₂³	C₄×C₇⋊C₈	1	2	yes	—
215	C₂³	C₄⋊(C₇⋊C₈)	1	2	yes	—
216	C₂³	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
217	C₂³	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
218	C₂³	C₂²×C₇⋊C₈	1	1	yes	characteristic
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	C₄×C₇⋊C₈	1	2	yes	—
224	C₂×C₄	#6377	1	2	yes	—
225	C₂×C₄	#6377	1	2	yes	—
226	C₂×C₄	C₄⋊(C₇⋊C₈)	1	2	yes	—
227	C₂×C₄	C₄⋊(C₇⋊C₈)	1	2	yes	—
228	C₂×C₄	C₄⋊(C₇⋊C₈)	1	2	yes	—
229	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	2	yes	—
230	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	2	yes	—
231	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
232	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
233	C₂×C₄	C₇⋊#95	1	2	yes	—
234	C₂×C₄	C₇⋊#95	1	2	yes	—
235	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
236	C₂×C₄	C₂×#1136	1	1	yes	characteristic
237	C₂×C₄	C₂×#1136	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
241	C₈	—	2	4	no	—
242	C₈	C₇⋊#95	1	4	yes	—
243	C₈	C₂×Dic₇⋊C₄	1	2	yes	—
244	C₈	C₂×(C₂×D₇)⋊C₄	1	2	yes	—
245	C₇	C₂×#1377	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	2	4	no	—
250	C₂²	#89319	1	2	yes	—
251	C₂²	#89319	1	2	yes	—
252	C₂²	#89348	1	4	yes	—
253	C₂²	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
254	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
255	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
256	C₄	—	4	4	no	—
257	C₄	#89319	1	2	yes	—
258	C₄	#89319	1	2	yes	—
259	C₄	C₂×#6377	1	1	yes	characteristic
260	C₄	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
261	C₄	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
262	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	2	no	—
266	C₂	#11725470	1	4	yes	direct product factor
267	C₂	C₂×#89319	1	1	yes	characteristic
268	C₂	C₂×#89348	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₇
commutator	C₂²×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂²×C₇×#97	C₂
socle	C₂³×C₇	C₂²⋊C₈
prime radical	C₂⁴×C₇	C₂×C₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂²×#97	D₇
hypocenter	C₇	C₂×#1377
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #11725470	C₂	direct (N × H)
13. C₂×#6386	C₄	semidirect
41. #6386	C₂×C₄	semidirect
210. C₂×C₇	#1377	semidirect
245. C₇	C₂×#1377	semidirect
266. C₂	#11725470	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
#11725470	2
C₂×#1377	7

Subgroups — C2×#11725470 #13268655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11725470 #13268655