Subgroups — C₇×#3298 #11742607

Order 896.

420 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	C₇×D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#547	C₂	1	1	yes	maximal, characteristic
3	C₇×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#483	C₂	1	2	yes	maximal, complemented
5	C₇×#475	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#465	C₂	1	2	yes	maximal, complemented
7	C₇×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#448	C₂	1	2	yes	maximal, complemented
9	C₄×C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×#103	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×#41	—	2	1	no	—
13	C₂²×C₇×Q₈	C₂²	1	2	yes	—
14	C₂×C₇×SD₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₇×SD₁₆	C₂²	1	2	yes	—
16	C₂²×C₇×C₈	C₂²	1	2	yes	—
17	C₇×C₄⋊Q₈	—	2	1	no	—
18	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₇×#123	—	2	2	no	—
21	C₇×C₂²⋊Q₈	—	2	1	no	—
22	C₇×C₂²⋊Q₈	—	2	2	no	—
23	C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₇×#121	C₂²	1	2	yes	—
25	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
26	C₄×C₇×D₄	—	2	1	no	—
27	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
28	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
29	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
30	C₇×#106	C₂²	1	2	yes	—
31	C₇×#106	C₂²	1	1	yes	characteristic
32	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₇×#103	C₂²	1	2	yes	—
34	C₇×#103	C₂²	1	2	yes	—
35	C₇×#103	C₂²	1	2	yes	—
36	C₇×#103	C₂²	1	2	yes	—
37	C₇×#103	C₂²	1	1	yes	characteristic
38	C₇×#102	C₂²	1	1	yes	characteristic
39	C₇×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
41	#3298	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₇×#41	—	4	4	no	—
43	C₂×C₇×Q₈	—	2	2	no	—
44	C₂×C₇×Q₈	—	2	1	no	—
45	C₂×C₇×Q₈	—	2	4	no	—
46	C₂×C₇×Q₈	—	4	2	no	—
47	C₂×C₇×Q₈	C₂³	1	2	yes	—
48	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
49	C₂×C₇×D₄	—	2	2	no	—
50	C₂×C₇×D₄	D₄	1	1	yes	characteristic
51	C₂×C₇×D₄	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₇	—	2	2	no	—
53	C₂²×C₄×C₇	—	2	2	no	—
54	C₂²×C₄×C₇	—	2	1	no	—
55	C₂²×C₄×C₇	C₂³	1	2	yes	—
56	C₇×SD₁₆	—	2	2	no	—
57	C₇×SD₁₆	—	4	2	no	—
58	C₂×C₇×C₈	—	4	2	no	—
59	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
61	C₂×C₇×C₈	C₂³	1	2	yes	—
62	C₇×C₄⋊C₄	—	2	2	no	—
63	C₇×C₄⋊C₄	—	2	1	no	—
64	C₇×C₄⋊C₄	—	2	1	no	—
65	C₇×C₄⋊C₄	—	2	1	no	—
66	C₇×C₄⋊C₄	—	2	2	no	—
67	C₇×C₄⋊C₄	—	2	1	no	—
68	C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
69	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
71	C₇×C₄⋊C₄	C₂³	1	2	yes	—
72	C₇×C₂²⋊C₄	—	2	2	no	—
73	C₇×C₂²⋊C₄	—	2	1	no	—
74	C₇×C₂²⋊C₄	—	2	2	no	—
75	C₇×C₂²⋊C₄	D₄	1	2	yes	—
76	C₄²×C₇	—	2	1	no	—
77	C₄²×C₇	C₂³	1	1	yes	characteristic
78	D₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
79	#547	C₂×C₇	1	1	yes	characteristic
80	#487	C₂×C₇	1	1	yes	characteristic, complemented
81	#483	C₂×C₇	1	2	yes	complemented
82	#475	C₂×C₇	1	1	yes	characteristic, complemented
83	#465	C₂×C₇	1	2	yes	complemented
84	#461	C₂×C₇	1	1	yes	characteristic, complemented
85	#448	C₂×C₇	1	2	yes	complemented
86	C₄×SD₁₆	C₂×C₇	1	1	yes	characteristic, complemented
87	C₈×D₄	C₂×C₇	1	1	yes	characteristic, complemented
88	C₂×#103	C₂×C₇	1	2	yes	complemented
89	C₂³×C₇	—	2	1	no	—
90	C₂³×C₇	C₂×D₄	1	2	yes	—
91	C₇×Q₈	—	2	1	no	—
92	C₇×Q₈	—	2	2	no	—
93	C₇×Q₈	—	2	2	no	—
94	C₇×Q₈	—	4	4	no	—
95	C₇×Q₈	—	4	2	no	—
96	C₇×Q₈	#41	1	2	yes	complemented
97	C₇×D₄	—	2	1	no	—
98	C₇×D₄	—	2	1	no	—
99	C₇×D₄	—	4	4	no	—
100	C₇×D₄	SD₁₆	1	4	yes	complemented
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	4	4	no	—
109	C₂×C₄×C₇	—	4	4	no	—
110	C₂×C₄×C₇	—	4	2	no	—
111	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
115	C₇×C₈	—	2	1	no	—
116	C₇×C₈	—	2	1	no	—
117	C₇×C₈	—	2	2	no	—
118	C₂×#41	—	2	1	no	—
119	C₂²×Q₈	C₂²×C₇	1	2	yes	—
120	C₂×SD₁₆	C₂²×C₇	1	1	yes	characteristic
121	C₂×SD₁₆	C₂²×C₇	1	2	yes	—
122	C₂²×C₈	C₂²×C₇	1	2	yes	—
123	C₄⋊Q₈	—	2	1	no	—
124	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
125	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
126	#123	—	2	2	no	—
127	C₂²⋊Q₈	—	2	1	no	—
128	C₂²⋊Q₈	—	2	2	no	—
129	C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
130	#121	C₂²×C₇	1	2	yes	—
131	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
132	C₄×D₄	—	2	1	no	—
133	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
134	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
136	#106	C₂²×C₇	1	1	yes	characteristic
137	#106	C₂²×C₇	1	2	yes	—
138	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
139	#103	C₂²×C₇	1	1	yes	characteristic
140	#103	C₂²×C₇	1	2	yes	—
141	#103	C₂²×C₇	1	2	yes	—
142	#103	C₂²×C₇	1	2	yes	—
143	#103	C₂²×C₇	1	2	yes	—
144	#102	C₂²×C₇	1	1	yes	characteristic
145	C₂²⋊C₈	C₂²×C₇	1	2	yes	—
146	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
147	C₂²×C₇	—	2	2	no	—
148	C₂²×C₇	—	2	2	no	—
149	C₂²×C₇	—	4	1	no	—
150	C₂²×C₇	—	4	1	no	—
151	C₂²×C₇	—	4	1	no	—
152	C₂²×C₇	C₂×SD₁₆	1	4	yes	—
153	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
154	C₄×C₇	—	2	2	no	—
155	C₄×C₇	—	2	2	no	—
156	C₄×C₇	—	2	1	no	—
157	C₄×C₇	—	4	2	no	—
158	C₄×C₇	—	4	1	no	—
159	C₄×C₇	—	4	2	no	—
160	C₄×C₇	—	4	1	no	—
161	C₄×C₇	C₂×SD₁₆	1	2	yes	—
162	C₄×C₇	C₂×#41	1	1	yes	characteristic
163	C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
164	#41	—	4	4	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	C₂³×C₇	1	2	yes	—
170	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	C₇×D₄	1	1	yes	characteristic
173	C₂×D₄	C₂³×C₇	1	1	yes	characteristic
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	C₂³×C₇	1	2	yes	—
178	SD₁₆	—	2	2	no	—
179	SD₁₆	—	4	2	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
182	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
183	C₂×C₈	C₂³×C₇	1	2	yes	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
191	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
192	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
193	C₄⋊C₄	C₂³×C₇	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	1	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	C₇×D₄	1	2	yes	—
198	C₄²	—	2	1	no	—
199	C₄²	C₂³×C₇	1	1	yes	characteristic
200	C₂×C₇	—	2	4	no	—
201	C₂×C₇	—	8	1	no	—
202	C₂×C₇	#546	1	1	yes	characteristic
203	C₂×C₇	C₂²×SD₁₆	1	1	yes	characteristic
204	C₂×C₇	#578	1	1	yes	characteristic
205	C₂³	—	2	1	no	—
206	C₂³	C₂×C₇×D₄	1	2	yes	—
207	Q₈	—	2	1	no	—
208	Q₈	—	2	2	no	—
209	Q₈	—	2	2	no	—
210	Q₈	—	4	4	no	—
211	Q₈	—	4	2	no	—
212	Q₈	C₇×#41	1	2	yes	complemented
213	D₄	—	2	1	no	—
214	D₄	—	2	1	no	—
215	D₄	—	4	4	no	—
216	D₄	C₇×SD₁₆	1	4	yes	complemented
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
228	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
229	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
230	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	2	no	—
234	C₇	#3298	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	C₂×C₇×SD₁₆	1	4	yes	—
241	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	1	no	—
249	C₄	C₂×C₇×SD₁₆	1	2	yes	—
250	C₄	C₂×C₇×#41	1	1	yes	characteristic
251	C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
252	C₂	—	2	4	no	—
253	C₂	—	8	1	no	—
254	C₂	C₇×#546	1	1	yes	characteristic
255	C₂	C₂²×C₇×SD₁₆	1	1	yes	characteristic
256	C₂	C₇×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
prime radical	C₇×#547	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×D₄×Q₈	C₂	semidirect
3. C₇×#487	C₂	semidirect
4. C₇×#483	C₂	semidirect
5. C₇×#475	C₂	semidirect
6. C₇×#465	C₂	semidirect
7. C₇×#461	C₂	semidirect
8. C₇×#448	C₂	semidirect
9. C₄×C₇×SD₁₆	C₂	semidirect
10. C₇×C₈×D₄	C₂	semidirect
11. C₂×C₇×#103	C₂	semidirect
41. #3298	C₇	direct (N × H)
78. D₄×Q₈	C₂×C₇	semidirect
80. #487	C₂×C₇	semidirect
81. #483	C₂×C₇	semidirect
82. #475	C₂×C₇	semidirect
83. #465	C₂×C₇	semidirect
84. #461	C₂×C₇	semidirect
85. #448	C₂×C₇	semidirect
86. C₄×SD₁₆	C₂×C₇	semidirect
87. C₈×D₄	C₂×C₇	semidirect
88. C₂×#103	C₂×C₇	semidirect
96. C₇×Q₈	#41	semidirect
100. C₇×D₄	SD₁₆	semidirect
212. Q₈	C₇×#41	semidirect
216. D₄	C₇×SD₁₆	semidirect
234. C₇	#3298	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×D₄×Q₈	2
C₇×#547	2
C₇×#487	2
C₇×#483	2
C₇×#475	2
C₇×#465	2
C₇×#461	2
C₇×#448	2
C₄×C₇×SD₁₆	2
C₇×C₈×D₄	2
C₂×C₇×#103	2
#3298	7

Subgroups — C7×#3298 #11742607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3298 #11742607