Subgroups — C₇×#21323 #13198539

Order 1792.

1182 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#382	C₂	1	1	yes	maximal, characteristic
4	C₇×#1993	C₂	1	4	yes	maximal, complemented
5	C₇×#1991	C₂	1	4	yes	maximal, complemented
6	C₇×#1736	C₂	1	4	yes	maximal
7	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
8	C₇×#533	—	2	4	no	—
9	C₇×#533	C₄	1	8	yes	complemented
10	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₇×#125	C₂²	1	2	yes	—
12	C₂×C₇×C₂²⋊Q₈	—	2	2	no	—
13	C₂×C₄×C₇×Q₈	—	2	2	no	—
14	C₂×C₇×#117	C₂²	1	2	yes	—
15	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
17	C₄×C₇×M₁₆	—	2	4	no	—
18	C₂×C₇×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
20	C₇×#382	—	2	2	no	—
21	C₇×#382	C₂²	1	2	yes	—
22	C₇×#382	C₂²	1	2	yes	—
23	C₂×C₇×#95	—	2	2	no	—
24	C₇×#340	C₂²	1	16	yes	—
25	#21323	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₇×Q₈	—	2	2	no	—
27	C₂³×C₄×C₇	—	2	2	no	—
28	C₂³×C₄×C₇	D₄	1	2	yes	—
29	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₂×C₇×M₁₆	—	2	4	no	—
31	C₂×C₇×M₁₆	—	2	4	no	—
32	C₂×C₇×M₁₆	Q₈	1	8	yes	complemented
33	C₂²×C₇×C₈	C₂³	1	4	yes	—
34	C₇×C₄⋊Q₈	—	2	4	no	—
35	C₇×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
36	C₇×#125	—	2	4	no	—
37	C₇×#125	C₂×C₄	1	8	yes	complemented
38	C₇×C₂²⋊Q₈	—	2	16	no	—
39	C₄×C₇×Q₈	—	2	16	no	—
40	C₇×#117	—	2	4	no	—
41	C₇×#117	C₂×C₄	1	8	yes	—
42	C₂×C₇×C₄⋊C₄	—	2	4	no	—
43	C₂×C₇×C₄⋊C₄	—	2	2	no	—
44	C₂×C₇×C₄⋊C₄	—	2	8	no	—
45	C₂×C₇×C₄⋊C₄	—	2	2	no	—
46	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
47	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
48	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
49	C₂×C₄²×C₇	—	2	2	no	—
50	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₇	D₄	1	2	yes	—
52	C₂×C₄²×C₇	D₄	1	4	yes	—
53	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₇×#97	—	2	4	no	—
56	C₄×C₇×C₈	—	2	4	no	—
57	C₇×#95	—	2	4	no	—
58	C₇×#95	—	2	4	no	—
59	C₂×#533	C₂×C₇	1	1	yes	characteristic
60	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic
61	C₂×#382	C₂×C₇	1	1	yes	characteristic
62	#1993	C₂×C₇	1	4	yes	complemented
63	#1991	C₂×C₇	1	4	yes	complemented
64	#1736	C₂×C₇	1	4	yes	—
65	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₇×Q₈	—	2	8	no	—
67	C₂×C₇×Q₈	—	4	8	no	—
68	C₂²×C₄×C₇	—	2	4	no	—
69	C₂²×C₄×C₇	—	2	4	no	—
70	C₂²×C₄×C₇	—	4	4	no	—
71	C₂²×C₄×C₇	—	4	8	no	—
72	C₂²×C₄×C₇	—	4	4	no	—
73	C₂²×C₄×C₇	C₂²⋊C₄	1	8	yes	—
74	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₇	#41	1	4	yes	—
81	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
82	C₇×M₁₆	—	2	16	no	—
83	C₂×C₇×C₈	—	2	4	no	—
84	C₂×C₇×C₈	—	2	4	no	—
85	C₂×C₇×C₈	C₂×Q₈	1	8	yes	—
86	C₇×C₄⋊C₄	—	2	16	no	—
87	C₇×C₄⋊C₄	—	2	8	no	—
88	C₇×C₄⋊C₄	—	2	4	no	—
89	C₇×C₄⋊C₄	—	2	8	no	—
90	C₇×C₄⋊C₄	—	4	4	no	—
91	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
92	C₇×C₂²⋊C₄	—	2	8	no	—
93	C₄²×C₇	—	2	8	no	—
94	C₄²×C₇	C₂²⋊C₄	1	8	yes	—
95	C₄²×C₇	C₂²×C₄	1	2	yes	—
96	C₄²×C₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₇	C₂×D₄	1	2	yes	—
98	C₄²×C₇	C₂×D₄	1	2	yes	—
99	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
100	#533	—	2	4	no	—
101	#533	C₄×C₇	1	8	yes	complemented
102	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
103	C₂×#125	C₂²×C₇	1	2	yes	—
104	C₂×C₂²⋊Q₈	—	2	2	no	—
105	C₂×C₄×Q₈	—	2	2	no	—
106	C₂×#117	C₂²×C₇	1	2	yes	—
107	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
108	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
109	C₄×M₁₆	—	2	4	no	—
110	C₂×#97	C₂²×C₇	1	2	yes	—
111	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
112	#382	—	2	2	no	—
113	#382	C₂²×C₇	1	2	yes	—
114	#382	C₂²×C₇	1	2	yes	—
115	C₂×#95	—	2	2	no	—
116	#340	C₂²×C₇	1	16	yes	—
117	C₂³×C₇	—	2	4	no	—
118	C₂³×C₇	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂³×C₇	#124	1	2	yes	—
120	C₂³×C₇	C₄⋊Q₈	1	2	yes	—
121	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
122	C₇×Q₈	—	4	16	no	—
123	C₂×C₄×C₇	—	2	4	no	—
124	C₂×C₄×C₇	—	2	4	no	—
125	C₂×C₄×C₇	—	2	8	no	—
126	C₂×C₄×C₇	—	4	8	no	—
127	C₂×C₄×C₇	—	4	8	no	—
128	C₂×C₄×C₇	—	4	4	no	—
129	C₂×C₄×C₇	—	4	8	no	—
130	C₂×C₄×C₇	—	8	8	no	—
131	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
132	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
133	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂×C₄×C₇	C₄×Q₈	1	8	yes	—
135	C₂×C₄×C₇	C₂²⋊Q₈	1	8	yes	—
136	C₂×C₄×C₇	#124	1	2	yes	—
137	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
138	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
141	C₂×C₄×C₇	C₂×#41	1	2	yes	—
142	C₇×C₈	—	2	8	no	—
143	C₂²×Q₈	—	2	2	no	—
144	C₂³×C₄	—	2	2	no	—
145	C₂³×C₄	C₇×D₄	1	2	yes	—
146	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
147	C₂×M₁₆	—	2	4	no	—
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	C₇×Q₈	1	8	yes	complemented
150	C₂²×C₈	C₂³×C₇	1	4	yes	—
151	C₄⋊Q₈	—	2	4	no	—
152	C₄⋊Q₈	C₂×C₄×C₇	1	8	yes	complemented
153	#125	—	2	4	no	—
154	#125	C₂×C₄×C₇	1	8	yes	complemented
155	C₂²⋊Q₈	—	2	16	no	—
156	C₄×Q₈	—	2	16	no	—
157	#117	—	2	4	no	—
158	#117	C₂×C₄×C₇	1	8	yes	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	8	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
164	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
165	C₂×C₂²⋊C₄	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
168	C₂×C₄²	C₇×D₄	1	2	yes	—
169	C₂×C₄²	C₇×D₄	1	4	yes	—
170	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
171	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
172	#97	—	2	4	no	—
173	C₄×C₈	—	2	4	no	—
174	#95	—	2	4	no	—
175	#95	—	2	4	no	—
176	C₂²×C₇	—	2	4	no	—
177	C₂²×C₇	—	2	4	no	—
178	C₂²×C₇	—	2	4	no	—
179	C₂²×C₇	#391	1	4	yes	—
180	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²×C₇	C₂×#124	1	1	yes	characteristic
182	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
183	C₂²×C₇	#575	1	4	yes	—
184	C₄×C₇	—	2	4	no	—
185	C₄×C₇	—	2	4	no	—
186	C₄×C₇	—	4	8	no	—
187	C₄×C₇	—	8	4	no	—
188	C₄×C₇	#391	1	4	yes	—
189	C₄×C₇	C₂×C₄×Q₈	1	2	yes	—
190	C₄×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
191	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
192	C₂×Q₈	—	2	8	no	—
193	C₂×Q₈	—	4	8	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	4	4	no	—
197	C₂²×C₄	—	4	8	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	C₇×C₂²⋊C₄	1	8	yes	—
200	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
201	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
202	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
203	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
205	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
206	C₂²×C₄	C₇×#41	1	4	yes	—
207	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
208	M₁₆	—	2	16	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	C₂×C₇×Q₈	1	8	yes	—
212	C₄⋊C₄	—	2	16	no	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
218	C₂²⋊C₄	—	2	8	no	—
219	C₄²	—	2	8	no	—
220	C₄²	C₇×C₂²⋊C₄	1	8	yes	—
221	C₄²	C₂²×C₄×C₇	1	2	yes	—
222	C₄²	C₂²×C₄×C₇	1	2	yes	—
223	C₄²	C₂×C₇×D₄	1	2	yes	—
224	C₄²	C₂×C₇×D₄	1	2	yes	—
225	C₂×C₇	—	2	4	no	—
226	C₂×C₇	C₂×#391	1	1	yes	characteristic
227	C₂×C₇	#2896	1	2	yes	—
228	C₂×C₇	#3140	1	2	yes	—
229	C₂×C₇	#3166	1	2	yes	—
230	C₂³	—	2	4	no	—
231	C₂³	C₂×C₇×C₂²⋊C₄	1	2	yes	—
232	C₂³	C₇×#124	1	2	yes	—
233	C₂³	C₇×C₄⋊Q₈	1	2	yes	—
234	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
235	Q₈	—	4	16	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	4	8	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	8	8	no	—
244	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
245	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
246	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
247	C₂×C₄	C₄×C₇×Q₈	1	8	yes	—
248	C₂×C₄	C₇×C₂²⋊Q₈	1	8	yes	—
249	C₂×C₄	C₇×#124	1	2	yes	—
250	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
251	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
254	C₂×C₄	C₂×C₇×#41	1	2	yes	—
255	C₈	—	2	8	no	—
256	C₇	#21323	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²	—	2	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	4	no	—
260	C₂²	C₇×#391	1	4	yes	—
261	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
262	C₂²	C₂×C₇×#124	1	1	yes	characteristic
263	C₂²	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
264	C₂²	C₇×#575	1	4	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	4	no	—
267	C₄	—	4	8	no	—
268	C₄	—	8	4	no	—
269	C₄	C₇×#391	1	4	yes	—
270	C₄	C₂×C₄×C₇×Q₈	1	2	yes	—
271	C₄	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	C₂×C₇×#391	1	1	yes	characteristic
274	C₂	C₇×#2896	1	2	yes	—
275	C₂	C₇×#3140	1	2	yes	—
276	C₂	C₇×#3166	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1993	C₂	semidirect
5. C₇×#1991	C₂	semidirect
9. C₇×#533	C₄	semidirect
25. #21323	C₇	direct (N × H)
32. C₂×C₇×M₁₆	Q₈	semidirect
35. C₇×C₄⋊Q₈	C₂×C₄	semidirect
37. C₇×#125	C₂×C₄	semidirect
62. #1993	C₂×C₇	semidirect
63. #1991	C₂×C₇	semidirect
101. #533	C₄×C₇	semidirect
149. C₂×M₁₆	C₇×Q₈	semidirect
152. C₄⋊Q₈	C₂×C₄×C₇	semidirect
154. #125	C₂×C₄×C₇	semidirect
256. C₇	#21323	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#533	2
C₂×C₄×C₇×M₁₆	2
C₂×C₇×#382	2
C₇×#1993	2
C₇×#1991	2
C₇×#1736	2
#21323	7

Subgroups — C7×#21323 #13198539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21323 #13198539