Subgroups — C₇×#33692 #13210908

Order 1792.

536 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3176	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#2955	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2238	C₂	1	2	yes	maximal, complemented
4	C₇×#2231	C₂	1	2	yes	maximal, complemented
5	C₇×#2213	C₂	1	2	yes	maximal, complemented
6	C₇×#2190	C₂	1	2	yes	maximal, complemented
7	C₇×AGL(1,ℤ/16)	C₂	1	4	yes	maximal, complemented
8	C₇×#2166	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#575	C₂²	1	1	yes	characteristic
10	C₂×C₇×#131	—	2	1	no	—
11	C₇×#540	—	2	1	no	—
12	C₇×#509	C₂²	1	4	yes	—
13	C₇×#499	C₂²	1	1	yes	characteristic, complemented
14	C₇×#493	C₂²	1	1	yes	characteristic, complemented
15	C₇×#490	—	2	1	no	—
16	C₇×#469	C₂²	1	2	yes	—
17	C₇×#466	—	2	1	no	—
18	C₇×#444	—	2	1	no	—
19	C₇×#444	C₂²	1	2	yes	—
20	C₇×#443	—	2	1	no	—
21	C₇×#443	C₂²	1	2	yes	—
22	C₇×#442	C₂²	1	2	yes	complemented
23	C₄×C₇×D₈	—	2	1	no	—
24	C₇×#430	C₂²	1	1	yes	characteristic
25	C₇×#421	—	2	1	no	—
26	C₇×#405	C₂²	1	1	yes	characteristic
27	C₇×#365	C₂²	1	4	yes	complemented
28	C₇×#364	C₂²	1	2	yes	complemented
29	C₇×#361	C₂²	1	4	yes	complemented
30	C₇×#360	C₂²	1	4	yes	—
31	C₇×#349	C₂²	1	2	yes	—
32	C₇×#347	C₂²	1	2	yes	complemented
33	#33692	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₇×#41	—	2	1	no	—
35	C₇×#137	—	2	2	no	—
36	C₇×AGL(1,ℤ/8)	—	2	2	no	—
37	C₇×#135	—	2	1	no	—
38	C₇×#135	C₂³	1	2	yes	—
39	C₂×C₇×D₈	—	2	1	no	—
40	C₂×C₇×D₈	C₂³	1	2	yes	—
41	C₇×#131	—	2	2	no	—
42	C₇×#131	—	2	2	no	—
43	C₇×#131	—	4	1	no	—
44	C₇×#131	—	4	1	no	—
45	C₂×C₇×M₁₆	—	2	1	no	—
46	C₂×C₇×M₁₆	—	4	1	no	—
47	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
48	C₂²×C₇×C₈	—	4	1	no	—
49	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
50	C₇×#127	—	2	1	no	—
51	C₇×#125	—	2	1	no	—
52	C₇×#123	—	4	2	no	—
53	C₇×#121	—	2	2	no	—
54	C₇×#121	—	4	2	no	—
55	C₄×C₇×D₄	—	2	2	no	—
56	C₇×#117	D₄	1	1	yes	characteristic
57	C₇×SD₃₂	—	2	4	no	—
58	C₇×D₁₆	—	2	4	no	—
59	C₇×M₃₂	C₂³	1	4	yes	—
60	C₇×#108	—	2	1	no	—
61	C₇×#108	C₂³	1	2	yes	—
62	C₇×#107	—	2	1	no	—
63	C₇×#106	C₂³	1	2	yes	—
64	C₇×#104	—	2	2	no	—
65	C₇×#104	—	2	2	no	—
66	C₇×#102	—	2	2	no	—
67	C₇×#102	—	2	2	no	—
68	C₇×#97	D₄	1	1	yes	characteristic
69	C₇×#97	C₂³	1	1	yes	characteristic
70	C₄×C₇×C₈	D₄	1	1	yes	characteristic
71	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
72	#3176	C₂×C₇	1	1	yes	characteristic, complemented
73	#2955	C₂×C₇	1	1	yes	characteristic, complemented
74	#2238	C₂×C₇	1	2	yes	complemented
75	#2231	C₂×C₇	1	2	yes	complemented
76	#2213	C₂×C₇	1	2	yes	complemented
77	#2190	C₂×C₇	1	2	yes	complemented
78	AGL(1,ℤ/16)	C₂×C₇	1	4	yes	complemented
79	#2166	C₂×C₇	1	1	yes	characteristic, complemented
80	C₇×#41	—	2	2	no	—
81	C₇×#41	—	2	2	no	—
82	C₇×#41	—	4	1	no	—
83	C₇×#41	—	4	1	no	—
84	C₂×C₇×Q₈	—	2	1	no	—
85	C₂×C₇×D₄	—	2	1	no	—
86	C₂×C₇×D₄	—	2	2	no	—
87	C₂×C₇×D₄	—	4	2	no	—
88	C₂×C₇×D₄	—	4	1	no	—
89	C₂×C₇×D₄	—	4	2	no	—
90	C₂²×C₄×C₇	—	4	1	no	—
91	C₂²×C₄×C₇	—	4	2	no	—
92	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
93	C₇×Q₁₆	—	2	1	no	—
94	C₇×Q₁₆	#41	1	2	yes	—
95	C₇×SD₁₆	—	2	2	no	—
96	C₇×SD₁₆	—	2	2	no	—
97	C₇×D₈	—	2	1	no	—
98	C₇×D₈	—	2	4	no	—
99	C₇×D₈	—	4	1	no	—
100	C₇×D₈	—	4	1	no	—
101	C₇×D₈	#41	1	2	yes	—
102	C₇×M₁₆	—	2	2	no	—
103	C₇×M₁₆	—	2	2	no	—
104	C₇×M₁₆	—	4	2	no	—
105	C₇×M₁₆	—	4	1	no	—
106	C₇×M₁₆	—	4	2	no	—
107	C₂×C₇×C₈	—	2	1	no	—
108	C₂×C₇×C₈	—	2	1	no	—
109	C₂×C₇×C₈	—	4	2	no	—
110	C₂×C₇×C₈	—	4	1	no	—
111	C₂×C₇×C₈	—	4	2	no	—
112	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
116	C₇×C₄⋊C₄	—	2	1	no	—
117	C₇×C₄⋊C₄	—	2	2	no	—
118	C₇×C₄⋊C₄	—	4	2	no	—
119	C₇×C₂²⋊C₄	—	2	1	no	—
120	C₇×C₂²⋊C₄	—	4	2	no	—
121	C₇×C₂²⋊C₄	—	4	2	no	—
122	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
124	C₇×C₁₆	—	2	4	no	—
125	#575	C₂²×C₇	1	1	yes	characteristic
126	C₂×#131	—	2	1	no	—
127	#540	—	2	1	no	—
128	#509	C₂²×C₇	1	4	yes	—
129	#499	C₂²×C₇	1	1	yes	characteristic, complemented
130	#493	C₂²×C₇	1	1	yes	characteristic, complemented
131	#490	—	2	1	no	—
132	#469	C₂²×C₇	1	2	yes	—
133	#466	—	2	1	no	—
134	#444	—	2	1	no	—
135	#444	C₂²×C₇	1	2	yes	—
136	#443	—	2	1	no	—
137	#443	C₂²×C₇	1	2	yes	—
138	#442	C₂²×C₇	1	2	yes	complemented
139	C₄×D₈	—	2	1	no	—
140	#430	C₂²×C₇	1	1	yes	characteristic
141	#421	—	2	1	no	—
142	#405	C₂²×C₇	1	1	yes	characteristic
143	#365	C₂²×C₇	1	4	yes	complemented
144	#364	C₂²×C₇	1	2	yes	complemented
145	#361	C₂²×C₇	1	4	yes	complemented
146	#360	C₂²×C₇	1	4	yes	—
147	#349	C₂²×C₇	1	2	yes	—
148	#347	C₂²×C₇	1	2	yes	complemented
149	C₂³×C₇	—	4	1	no	—
150	C₂³×C₇	—	4	2	no	—
151	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₇×Q₈	—	2	2	no	—
153	C₇×Q₈	—	4	1	no	—
154	C₇×D₄	—	2	2	no	—
155	C₇×D₄	—	4	1	no	—
156	C₇×D₄	—	4	2	no	—
157	C₇×D₄	—	4	2	no	—
158	C₇×D₄	—	4	2	no	—
159	C₇×D₄	—	4	2	no	—
160	C₇×D₄	—	16	2	no	—
161	C₇×D₄	—	16	2	no	—
162	C₂×C₄×C₇	—	2	1	no	—
163	C₂×C₄×C₇	—	2	1	no	—
164	C₂×C₄×C₇	—	2	1	no	—
165	C₂×C₄×C₇	—	2	1	no	—
166	C₂×C₄×C₇	—	4	2	no	—
167	C₂×C₄×C₇	—	4	1	no	—
168	C₂×C₄×C₇	—	4	2	no	—
169	C₂×C₄×C₇	—	4	2	no	—
170	C₂×C₄×C₇	—	16	2	no	—
171	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
173	C₇×C₈	—	2	1	no	—
174	C₇×C₈	—	2	1	no	—
175	C₇×C₈	—	4	2	no	—
176	C₇×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₇×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₇×C₈	C₂×#41	1	2	yes	—
179	C₂×#41	—	2	1	no	—
180	#137	—	2	2	no	—
181	AGL(1,ℤ/8)	—	2	2	no	—
182	#135	—	2	1	no	—
183	#135	C₂³×C₇	1	2	yes	—
184	C₂×D₈	—	2	1	no	—
185	C₂×D₈	C₂³×C₇	1	2	yes	—
186	#131	—	2	2	no	—
187	#131	—	2	2	no	—
188	#131	—	4	1	no	—
189	#131	—	4	1	no	—
190	C₂×M₁₆	—	2	1	no	—
191	C₂×M₁₆	—	4	1	no	—
192	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
193	C₂²×C₈	—	4	1	no	—
194	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
195	#127	—	2	1	no	—
196	#125	—	2	1	no	—
197	#123	—	4	2	no	—
198	#121	—	2	2	no	—
199	#121	—	4	2	no	—
200	C₄×D₄	—	2	2	no	—
201	#117	C₇×D₄	1	1	yes	characteristic
202	SD₃₂	—	2	4	no	—
203	D₁₆	—	2	4	no	—
204	M₃₂	C₂³×C₇	1	4	yes	—
205	#108	—	2	1	no	—
206	#108	C₂³×C₇	1	2	yes	—
207	#107	—	2	1	no	—
208	#106	C₂³×C₇	1	2	yes	—
209	#104	—	2	2	no	—
210	#104	—	2	2	no	—
211	#102	—	2	2	no	—
212	#102	—	2	2	no	—
213	#97	C₇×D₄	1	1	yes	characteristic
214	#97	C₂³×C₇	1	1	yes	characteristic
215	C₄×C₈	C₇×D₄	1	1	yes	characteristic
216	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
217	C₂²×C₇	—	2	1	no	—
218	C₂²×C₇	—	4	1	no	—
219	C₂²×C₇	—	4	2	no	—
220	C₂²×C₇	—	8	2	no	—
221	C₂²×C₇	—	8	2	no	—
222	C₂²×C₇	—	8	2	no	—
223	C₂²×C₇	—	8	2	no	—
224	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
225	C₄×C₇	—	2	1	no	—
226	C₄×C₇	—	4	1	no	—
227	C₄×C₇	—	4	1	no	—
228	C₄×C₇	—	4	2	no	—
229	C₄×C₇	—	8	2	no	—
230	C₄×C₇	#525	1	1	yes	characteristic
231	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	#41	—	4	1	no	—
235	#41	—	4	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
245	Q₁₆	—	2	1	no	—
246	Q₁₆	C₇×#41	1	2	yes	—
247	SD₁₆	—	2	2	no	—
248	SD₁₆	—	2	2	no	—
249	D₈	—	2	1	no	—
250	D₈	—	2	4	no	—
251	D₈	—	4	1	no	—
252	D₈	—	4	1	no	—
253	D₈	C₇×#41	1	2	yes	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	4	2	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	4	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
265	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
266	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
267	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
275	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
276	C₁₆	—	2	4	no	—
277	C₂×C₇	—	2	1	no	—
278	C₂×C₇	—	4	1	no	—
279	C₂×C₇	—	8	2	no	—
280	C₂×C₇	—	16	2	no	—
281	C₂×C₇	#3109	1	1	yes	characteristic
282	C₂³	—	4	1	no	—
283	C₂³	—	4	2	no	—
284	C₂³	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
285	Q₈	—	2	2	no	—
286	Q₈	—	4	1	no	—
287	D₄	—	2	2	no	—
288	D₄	—	4	1	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	16	2	no	—
294	D₄	—	16	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	16	2	no	—
304	C₂×C₄	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	4	2	no	—
309	C₈	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₈	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₈	C₂×C₇×#41	1	2	yes	—
312	C₇	#33692	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	2	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₇×#525	1	1	yes	characteristic
327	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	4	1	no	—
330	C₂	—	8	2	no	—
331	C₂	—	16	2	no	—
332	C₂	C₇×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3176	C₂	semidirect
2. C₇×#2955	C₂	semidirect
3. C₇×#2238	C₂	semidirect
4. C₇×#2231	C₂	semidirect
5. C₇×#2213	C₂	semidirect
6. C₇×#2190	C₂	semidirect
7. C₇×AGL(1,ℤ/16)	C₂	semidirect
8. C₇×#2166	C₂	semidirect
13. C₇×#499	C₂²	semidirect
14. C₇×#493	C₂²	semidirect
22. C₇×#442	C₂²	semidirect
27. C₇×#365	C₂²	semidirect
28. C₇×#364	C₂²	semidirect
29. C₇×#361	C₂²	semidirect
32. C₇×#347	C₂²	semidirect
33. #33692	C₇	direct (N × H)
72. #3176	C₂×C₇	semidirect
73. #2955	C₂×C₇	semidirect
74. #2238	C₂×C₇	semidirect
75. #2231	C₂×C₇	semidirect
76. #2213	C₂×C₇	semidirect
77. #2190	C₂×C₇	semidirect
78. AGL(1,ℤ/16)	C₂×C₇	semidirect
79. #2166	C₂×C₇	semidirect
129. #499	C₂²×C₇	semidirect
130. #493	C₂²×C₇	semidirect
138. #442	C₂²×C₇	semidirect
143. #365	C₂²×C₇	semidirect
144. #364	C₂²×C₇	semidirect
145. #361	C₂²×C₇	semidirect
148. #347	C₂²×C₇	semidirect
312. C₇	#33692	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×#3176	2
C₇×#2955	2
C₇×#2238	2
C₇×#2231	2
C₇×#2213	2
C₇×#2190	2
C₇×AGL(1,ℤ/16)	2
C₇×#2166	2
#33692	7

Subgroups — C7×#33692 #13210908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#33692 #13210908