Subgroups — C₇×#33705 #13210921

Order 1792.

578 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic
3	C₇×#2245	C₂	1	4	yes	maximal, complemented
4	C₇×#2241	C₂	1	4	yes	maximal, complemented
5	C₇×#2141	C₂	1	4	yes	maximal
6	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₇×Q₁₆	—	2	1	no	—
8	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
9	C₇×#533	—	2	1	no	—
10	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₇×M₃₂	C₂²	1	2	yes	—
12	C₇×#500	C₂²	1	2	yes	—
13	C₇×#499	—	2	1	no	—
14	C₇×#499	C₂²	1	2	yes	—
15	C₇×#498	—	2	1	no	—
16	C₇×#494	—	2	1	no	—
17	C₇×#494	—	2	1	no	—
18	C₇×#494	C₂²	1	2	yes	—
19	C₇×#494	C₂²	1	2	yes	—
20	C₇×#427	—	2	1	no	—
21	C₇×#427	C₂²	1	2	yes	—
22	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
23	C₇×#358	C₂²	1	16	yes	—
24	C₇×#346	C₂²	1	2	yes	—
25	C₄×C₇×C₁₆	C₂²	1	2	yes	—
26	#33705	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₇×Q₈	—	2	2	no	—
28	C₂×C₇×Q₁₆	—	2	2	no	—
29	C₂×C₇×Q₁₆	—	2	2	no	—
30	C₂×C₇×Q₁₆	—	2	2	no	—
31	C₂×C₇×Q₁₆	—	2	8	no	—
32	C₂×C₇×Q₁₆	—	4	4	no	—
33	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
34	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₇×C₄⋊Q₈	—	2	2	no	—
37	C₇×C₄⋊Q₈	—	2	2	no	—
38	C₇×C₄⋊Q₈	—	2	2	no	—
39	C₇×C₄⋊Q₈	—	4	2	no	—
40	C₇×#125	—	2	2	no	—
41	C₇×C₂²⋊Q₈	—	4	2	no	—
42	C₄×C₇×Q₈	—	4	2	no	—
43	C₇×#117	—	2	2	no	—
44	C₂×C₇×C₄⋊C₄	—	4	2	no	—
45	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
46	C₇×M₃₂	—	2	4	no	—
47	C₂×C₇×C₁₆	C₂³	1	4	yes	—
48	C₇×#107	—	2	2	no	—
49	C₇×#107	C₂³	1	4	yes	—
50	C₇×#106	—	2	2	no	—
51	C₇×#106	—	2	2	no	—
52	C₄×C₇×C₈	D₄	1	1	yes	characteristic
53	C₄×C₇×C₈	D₄	1	1	yes	characteristic
54	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
56	#3160	C₂×C₇	1	1	yes	characteristic
57	C₂×#494	C₂×C₇	1	1	yes	characteristic
58	#2245	C₂×C₇	1	4	yes	complemented
59	#2241	C₂×C₇	1	4	yes	complemented
60	#2141	C₂×C₇	1	4	yes	—
61	C₄×M₃₂	C₂×C₇	1	1	yes	characteristic
62	C₂×C₇×Q₈	—	2	4	no	—
63	C₂×C₇×Q₈	—	4	2	no	—
64	C₂×C₇×Q₈	—	4	2	no	—
65	C₂×C₇×Q₈	—	4	4	no	—
66	C₂×C₇×Q₈	—	4	4	no	—
67	C₂²×C₄×C₇	—	4	2	no	—
68	C₂²×C₄×C₇	D₈	1	2	yes	—
69	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
70	C₇×Q₁₆	—	2	8	no	—
71	C₇×Q₁₆	—	4	4	no	—
72	C₇×Q₁₆	—	4	4	no	—
73	C₇×Q₁₆	—	4	4	no	—
74	C₂×C₇×C₈	SD₁₆	1	4	yes	—
75	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₇×C₈	#41	1	4	yes	—
79	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
80	C₇×C₄⋊C₄	—	2	4	no	—
81	C₇×C₄⋊C₄	—	4	4	no	—
82	C₇×C₄⋊C₄	—	4	4	no	—
83	C₇×C₄⋊C₄	—	4	4	no	—
84	C₇×C₄⋊C₄	—	4	2	no	—
85	C₇×C₂²⋊C₄	—	4	2	no	—
86	C₄²×C₇	—	4	2	no	—
87	C₄²×C₇	D₈	1	2	yes	—
88	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
90	C₇×C₁₆	—	2	4	no	—
91	C₂²×Q₁₆	—	2	1	no	—
92	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
93	#533	—	2	1	no	—
94	C₂×C₄⋊Q₈	—	2	1	no	—
95	C₂×M₃₂	C₂²×C₇	1	2	yes	—
96	#500	C₂²×C₇	1	2	yes	—
97	#499	—	2	1	no	—
98	#499	C₂²×C₇	1	2	yes	—
99	#498	—	2	1	no	—
100	#494	—	2	1	no	—
101	#494	—	2	1	no	—
102	#494	C₂²×C₇	1	2	yes	—
103	#494	C₂²×C₇	1	2	yes	—
104	#427	—	2	1	no	—
105	#427	C₂²×C₇	1	2	yes	—
106	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
107	#358	C₂²×C₇	1	16	yes	—
108	#346	C₂²×C₇	1	2	yes	—
109	C₄×C₁₆	C₂²×C₇	1	2	yes	—
110	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
111	C₇×Q₈	—	4	4	no	—
112	C₇×Q₈	—	4	4	no	—
113	C₇×Q₈	—	8	4	no	—
114	C₇×Q₈	—	8	4	no	—
115	C₂×C₄×C₇	—	2	4	no	—
116	C₂×C₄×C₇	—	4	4	no	—
117	C₂×C₄×C₇	—	4	4	no	—
118	C₂×C₄×C₇	—	8	4	no	—
119	C₂×C₄×C₇	#124	1	2	yes	—
120	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
124	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
125	C₇×C₈	#124	1	2	yes	—
126	C₇×C₈	C₂×SD₁₆	1	2	yes	—
127	C₇×C₈	C₂×SD₁₆	1	2	yes	—
128	C₇×C₈	C₂×#41	1	2	yes	—
129	C₂²×Q₈	—	2	2	no	—
130	C₂×Q₁₆	—	2	2	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	8	no	—
134	C₂×Q₁₆	—	4	4	no	—
135	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
136	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
137	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
138	C₄⋊Q₈	—	2	2	no	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	4	2	no	—
142	#125	—	2	2	no	—
143	C₂²⋊Q₈	—	4	2	no	—
144	C₄×Q₈	—	4	2	no	—
145	#117	—	2	2	no	—
146	C₂×C₄⋊C₄	—	4	2	no	—
147	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
148	M₃₂	—	2	4	no	—
149	C₂×C₁₆	C₂³×C₇	1	4	yes	—
150	#107	—	2	2	no	—
151	#107	C₂³×C₇	1	4	yes	—
152	#106	—	2	2	no	—
153	#106	—	2	2	no	—
154	C₄×C₈	C₇×D₄	1	1	yes	characteristic
155	C₄×C₈	C₇×D₄	1	1	yes	characteristic
156	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
157	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
158	C₂²×C₇	—	2	2	no	—
159	C₂²×C₇	#486	1	2	yes	—
160	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₇	—	2	2	no	—
162	C₄×C₇	—	8	4	no	—
163	C₄×C₇	—	8	4	no	—
164	C₄×C₇	#486	1	2	yes	—
165	C₄×C₇	#510	1	4	yes	—
166	C₄×C₇	C₂×#124	1	1	yes	characteristic
167	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
168	C₂×Q₈	—	2	4	no	—
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₇×D₈	1	2	yes	—
175	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
176	Q₁₆	—	2	8	no	—
177	Q₁₆	—	4	4	no	—
178	Q₁₆	—	4	4	no	—
179	Q₁₆	—	4	4	no	—
180	C₂×C₈	C₇×SD₁₆	1	4	yes	—
181	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
182	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
183	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
184	C₂×C₈	C₇×#41	1	4	yes	—
185	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₂²⋊C₄	—	4	2	no	—
192	C₄²	—	4	2	no	—
193	C₄²	C₇×D₈	1	2	yes	—
194	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
195	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
196	C₁₆	—	2	4	no	—
197	C₂×C₇	—	2	2	no	—
198	C₂×C₇	C₂×#486	1	1	yes	characteristic
199	C₂×C₇	C₂×#510	1	2	yes	—
200	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
201	Q₈	—	4	4	no	—
202	Q₈	—	4	4	no	—
203	Q₈	—	8	4	no	—
204	Q₈	—	8	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	C₇×#124	1	2	yes	—
210	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
211	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
212	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
213	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
214	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
215	C₈	C₇×#124	1	2	yes	—
216	C₈	C₂×C₇×SD₁₆	1	2	yes	—
217	C₈	C₂×C₇×SD₁₆	1	2	yes	—
218	C₈	C₂×C₇×#41	1	2	yes	—
219	C₇	#33705	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²	—	2	2	no	—
221	C₂²	C₇×#486	1	2	yes	—
222	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	8	4	no	—
226	C₄	C₇×#486	1	2	yes	—
227	C₄	C₇×#510	1	4	yes	—
228	C₄	C₂×C₇×#124	1	1	yes	characteristic
229	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×C₇×#486	1	1	yes	characteristic
232	C₂	C₂×C₇×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
prime radical	C₂³×C₇	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#2245	C₂	semidirect
4. C₇×#2241	C₂	semidirect
26. #33705	C₇	direct (N × H)
58. #2245	C₂×C₇	semidirect
59. #2241	C₂×C₇	semidirect
219. C₇	#33705	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×#3160	2
C₂×C₇×#494	2
C₇×#2245	2
C₇×#2241	2
C₇×#2141	2
C₄×C₇×M₃₂	2
#33705	7

Subgroups — C7×#33705 #13210921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#33705 #13210921