Subgroups — C₇×#23198 #13200414

Order 1792.

946 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#492	C₂	1	1	yes	maximal, characteristic
2	C₇×#2949	C₂	1	4	yes	maximal, complemented
3	C₇×#1977	C₂	1	4	yes	maximal, complemented
4	C₇×#1887	C₂	1	2	yes	maximal
5	C₇×#1886	C₂	1	2	yes	maximal, complemented
6	C₇×#1772	C₂	1	2	yes	maximal, complemented
7	C₂²×C₇×SD₁₆	C₂²	1	4	yes	—
8	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₇×#127	C₂²	1	1	yes	characteristic
10	C₇×#519	C₂²	1	2	yes	—
11	C₇×#518	C₂²	1	2	yes	—
12	C₇×#492	C₄	1	8	yes	complemented
13	C₇×#440	—	2	8	no	—
14	C₄×C₇×SD₁₆	—	2	8	no	—
15	C₂×C₇×#108	C₂²	1	2	yes	—
16	C₇×#427	C₂²	1	2	yes	—
17	C₇×#419	C₂²	1	4	yes	—
18	C₇×#418	C₂²	1	4	yes	—
19	C₂×C₇×#101	C₂²	1	2	yes	—
20	C₂×C₇×#100	C₂²	1	2	yes	—
21	C₇×#405	C₂²	1	4	yes	—
22	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
23	C₇×#339	C₂²	1	4	yes	—
24	#23198	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₇×Q₈	C₂³	1	2	yes	—
26	C₂²×C₇×D₄	—	4	1	no	—
27	C₂²×C₇×D₄	C₂³	1	2	yes	—
28	C₂×C₇×SD₁₆	—	2	4	no	—
29	C₂×C₇×SD₁₆	—	2	4	no	—
30	C₂×C₇×SD₁₆	—	2	8	no	—
31	C₂×C₇×SD₁₆	C₂×C₄	1	16	yes	complemented
32	C₂×C₇×M₁₆	C₂³	1	4	yes	—
33	C₂²×C₇×C₈	C₂³	1	2	yes	—
34	C₇×C₄⋊Q₈	—	4	1	no	—
35	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
36	C₇×#127	—	4	1	no	—
37	C₇×#127	C₂×C₄	1	4	yes	—
38	C₄×C₇×Q₈	—	2	8	no	—
39	C₄×C₇×D₄	—	2	8	no	—
40	C₇×#117	—	4	2	no	—
41	C₇×#117	C₂³	1	4	yes	—
42	C₂×C₇×C₄⋊C₄	—	4	1	no	—
43	C₂×C₇×C₂²⋊C₄	—	4	2	no	—
44	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
45	C₇×#108	—	2	4	no	—
46	C₇×#107	D₄	1	4	yes	complemented
47	C₇×#106	D₄	1	4	yes	complemented
48	C₇×#103	—	2	8	no	—
49	C₇×#102	—	2	8	no	—
50	C₇×#101	—	4	2	no	—
51	C₇×#100	—	4	2	no	—
52	C₇×#97	—	2	4	no	—
53	C₄×C₇×C₈	—	2	4	no	—
54	C₄×C₇×C₈	C₂×C₄	1	4	yes	complemented
55	C₂×#492	C₂×C₇	1	1	yes	characteristic
56	#2949	C₂×C₇	1	4	yes	complemented
57	#1977	C₂×C₇	1	4	yes	complemented
58	#1887	C₂×C₇	1	2	yes	—
59	#1886	C₂×C₇	1	2	yes	complemented
60	#1772	C₂×C₇	1	2	yes	complemented
61	C₂⁴×C₇	—	2	2	no	—
62	C₂×C₇×Q₈	—	2	2	no	—
63	C₂×C₇×Q₈	—	2	4	no	—
64	C₂×C₇×Q₈	—	2	4	no	—
65	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
66	C₂×C₇×D₄	—	2	4	no	—
67	C₂×C₇×D₄	—	2	4	no	—
68	C₂×C₇×D₄	—	2	2	no	—
69	C₂×C₇×D₄	—	4	4	no	—
70	C₂×C₇×D₄	—	8	4	no	—
71	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₇	—	2	1	no	—
73	C₂²×C₄×C₇	—	2	2	no	—
74	C₂²×C₄×C₇	—	4	4	no	—
75	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
76	C₇×SD₁₆	—	2	16	no	—
77	C₇×SD₁₆	—	2	16	no	—
78	C₇×M₁₆	—	2	4	no	—
79	C₇×M₁₆	—	2	4	no	—
80	C₂×C₇×C₈	—	2	4	no	—
81	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
82	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
84	C₂×C₇×C₈	#41	1	4	yes	—
85	C₇×C₄⋊C₄	—	4	4	no	—
86	C₇×C₄⋊C₄	—	4	4	no	—
87	C₇×C₄⋊C₄	C₂×D₄	1	8	yes	—
88	C₇×C₂²⋊C₄	—	2	4	no	—
89	C₇×C₂²⋊C₄	—	4	4	no	—
90	C₄²×C₇	—	2	4	no	—
91	C₄²×C₇	—	2	1	no	—
92	C₄²×C₇	—	4	4	no	—
93	C₄²×C₇	C₂²×C₄	1	2	yes	—
94	C₂²×SD₁₆	C₂²×C₇	1	4	yes	—
95	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
96	C₂×#127	C₂²×C₇	1	1	yes	characteristic
97	#519	C₂²×C₇	1	2	yes	—
98	#518	C₂²×C₇	1	2	yes	—
99	#492	C₄×C₇	1	8	yes	complemented
100	#440	—	2	8	no	—
101	C₄×SD₁₆	—	2	8	no	—
102	C₂×#108	C₂²×C₇	1	2	yes	—
103	#427	C₂²×C₇	1	2	yes	—
104	#419	C₂²×C₇	1	4	yes	—
105	#418	C₂²×C₇	1	4	yes	—
106	C₂×#101	C₂²×C₇	1	2	yes	—
107	C₂×#100	C₂²×C₇	1	2	yes	—
108	#405	C₂²×C₇	1	4	yes	—
109	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
110	#339	C₂²×C₇	1	4	yes	—
111	C₂³×C₇	—	2	4	no	—
112	C₂³×C₇	—	4	2	no	—
113	C₂³×C₇	—	4	2	no	—
114	C₂³×C₇	—	8	4	no	—
115	C₂³×C₇	H(5,𝔽₂)	1	1	yes	characteristic
116	C₇×Q₈	—	2	8	no	—
117	C₇×Q₈	—	4	4	no	—
118	C₇×D₄	—	2	8	no	—
119	C₇×D₄	—	4	4	no	—
120	C₇×D₄	—	8	8	no	—
121	C₂×C₄×C₇	—	2	4	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	2	4	no	—
124	C₂×C₄×C₇	—	2	8	no	—
125	C₂×C₄×C₇	—	4	2	no	—
126	C₂×C₄×C₇	—	4	2	no	—
127	C₂×C₄×C₇	—	16	4	no	—
128	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₇	C₂×#41	1	2	yes	—
131	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
132	C₇×C₈	—	4	4	no	—
133	C₇×C₈	C₄×D₄	1	4	yes	—
134	C₇×C₈	C₄×D₄	1	4	yes	—
135	C₂²×Q₈	C₂³×C₇	1	2	yes	—
136	C₂²×D₄	—	4	1	no	—
137	C₂²×D₄	C₂³×C₇	1	2	yes	—
138	C₂×SD₁₆	—	2	4	no	—
139	C₂×SD₁₆	—	2	4	no	—
140	C₂×SD₁₆	—	2	8	no	—
141	C₂×SD₁₆	C₂×C₄×C₇	1	16	yes	complemented
142	C₂×M₁₆	C₂³×C₇	1	4	yes	—
143	C₂²×C₈	C₂³×C₇	1	2	yes	—
144	C₄⋊Q₈	—	4	1	no	—
145	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	complemented
146	#127	—	4	1	no	—
147	#127	C₂×C₄×C₇	1	4	yes	—
148	C₄×Q₈	—	2	8	no	—
149	C₄×D₄	—	2	8	no	—
150	#117	—	4	2	no	—
151	#117	C₂³×C₇	1	4	yes	—
152	C₂×C₄⋊C₄	—	4	1	no	—
153	C₂×C₂²⋊C₄	—	4	2	no	—
154	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
155	#108	—	2	4	no	—
156	#107	C₇×D₄	1	4	yes	complemented
157	#106	C₇×D₄	1	4	yes	complemented
158	#103	—	2	8	no	—
159	#102	—	2	8	no	—
160	#101	—	4	2	no	—
161	#100	—	4	2	no	—
162	#97	—	2	4	no	—
163	C₄×C₈	—	2	4	no	—
164	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
165	C₂²×C₇	—	2	2	no	—
166	C₂²×C₇	—	4	4	no	—
167	C₂²×C₇	—	8	4	no	—
168	C₂²×C₇	—	8	4	no	—
169	C₂²×C₇	—	16	2	no	—
170	C₂²×C₇	#518	1	1	yes	characteristic
171	C₂²×C₇	D₄²	1	1	yes	characteristic
172	C₂²×C₇	#559	1	1	yes	characteristic
173	C₄×C₇	—	2	4	no	—
174	C₄×C₇	—	4	4	no	—
175	C₄×C₇	—	4	8	no	—
176	C₄×C₇	C₂×C₄×D₄	1	2	yes	—
177	C₄×C₇	#550	1	2	yes	—
178	C₂⁴	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	4	4	no	—
187	C₂×D₄	—	8	4	no	—
188	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
193	SD₁₆	—	2	16	no	—
194	SD₁₆	—	2	16	no	—
195	M₁₆	—	2	4	no	—
196	M₁₆	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
199	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
200	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
201	C₂×C₈	C₇×#41	1	4	yes	—
202	C₄⋊C₄	—	4	4	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	C₂×C₇×D₄	1	8	yes	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	—	4	4	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	4	4	no	—
210	C₄²	C₂²×C₄×C₇	1	2	yes	—
211	C₂×C₇	—	2	1	no	—
212	C₂×C₇	—	2	1	no	—
213	C₂×C₇	—	8	4	no	—
214	C₂×C₇	#2366	1	1	yes	characteristic
215	C₂×C₇	#3291	1	2	yes	—
216	C₂³	—	2	4	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	2	no	—
219	C₂³	—	8	4	no	—
220	C₂³	C₇×H(5,𝔽₂)	1	1	yes	characteristic
221	Q₈	—	2	8	no	—
222	Q₈	—	4	4	no	—
223	D₄	—	2	8	no	—
224	D₄	—	4	4	no	—
225	D₄	—	8	8	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	16	4	no	—
233	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
234	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
235	C₂×C₄	C₂×C₇×#41	1	2	yes	—
236	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
237	C₈	—	4	4	no	—
238	C₈	C₄×C₇×D₄	1	4	yes	—
239	C₈	C₄×C₇×D₄	1	4	yes	—
240	C₇	#23198	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	—	8	4	no	—
244	C₂²	—	8	4	no	—
245	C₂²	—	16	2	no	—
246	C₂²	C₇×#518	1	1	yes	characteristic
247	C₂²	C₇×D₄²	1	1	yes	characteristic
248	C₂²	C₇×#559	1	1	yes	characteristic
249	C₄	—	2	4	no	—
250	C₄	—	4	8	no	—
251	C₄	—	4	4	no	—
252	C₄	C₂×C₄×C₇×D₄	1	2	yes	—
253	C₄	C₇×#550	1	2	yes	—
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	8	4	no	—
257	C₂	C₇×#2366	1	1	yes	characteristic
258	C₂	C₇×#3291	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2949	C₂	semidirect
3. C₇×#1977	C₂	semidirect
5. C₇×#1886	C₂	semidirect
6. C₇×#1772	C₂	semidirect
12. C₇×#492	C₄	semidirect
24. #23198	C₇	direct (N × H)
31. C₂×C₇×SD₁₆	C₂×C₄	semidirect
35. C₇×C₄⋊Q₈	C₂×C₄	semidirect
46. C₇×#107	D₄	semidirect
47. C₇×#106	D₄	semidirect
54. C₄×C₇×C₈	C₂×C₄	semidirect
56. #2949	C₂×C₇	semidirect
57. #1977	C₂×C₇	semidirect
59. #1886	C₂×C₇	semidirect
60. #1772	C₂×C₇	semidirect
99. #492	C₄×C₇	semidirect
141. C₂×SD₁₆	C₂×C₄×C₇	semidirect
145. C₄⋊Q₈	C₂×C₄×C₇	semidirect
156. #107	C₇×D₄	semidirect
157. #106	C₇×D₄	semidirect
164. C₄×C₈	C₂×C₄×C₇	semidirect
240. C₇	#23198	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×#492	2
C₇×#2949	2
C₇×#1977	2
C₇×#1887	2
C₇×#1886	2
C₇×#1772	2
#23198	7

Subgroups — C7×#23198 #13200414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23198 #13200414