Subgroups — C₇×#33857 #13211073

Order 1792.

646 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3416	C₂	1	2	yes	maximal, complemented
2	C₇×#3290	C₂	1	2	yes	maximal, complemented
3	C₇×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2223	C₂	1	2	yes	maximal, complemented
5	C₇×#2212	C₂	1	2	yes	maximal, complemented
6	C₇×#2195	C₂	1	2	yes	maximal, complemented
7	C₇×#2192	C₂	1	2	yes	maximal, complemented
8	C₇×#2178	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#2170	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#578	C₂²	1	2	yes	—
11	C₇×#577	—	2	2	no	—
12	C₇×#576	C₂²	1	2	yes	complemented
13	C₇×#510	—	2	2	no	—
14	C₇×#510	C₂²	1	2	yes	complemented
15	C₇×#509	—	2	2	no	—
16	C₇×#509	C₂²	1	2	yes	—
17	C₇×#508	—	2	2	no	—
18	C₇×#508	C₂²	1	1	yes	characteristic
19	C₂×C₇×Q₃₂	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₇×SD₃₂	—	2	2	no	—
21	C₂×C₇×SD₃₂	C₂²	1	2	yes	complemented
22	C₂×C₇×D₁₆	C₂²	1	1	yes	characteristic
23	C₇×#504	C₂²	1	2	yes	complemented
24	C₇×#495	C₂²	1	2	yes	complemented
25	C₇×#473	C₂²	1	2	yes	complemented
26	C₇×#472	C₂²	1	2	yes	—
27	C₇×#471	—	2	2	no	—
28	C₇×#455	—	2	2	no	—
29	C₇×#454	—	2	2	no	—
30	C₇×#443	C₂²	1	1	yes	characteristic
31	C₇×#443	C₂²	1	2	yes	complemented
32	C₇×#368	C₂²	1	1	yes	characteristic, complemented
33	C₇×#364	C₂²	1	1	yes	characteristic, complemented
34	C₇×#362	C₂²	1	2	yes	complemented
35	C₇×#361	C₂²	1	2	yes	complemented
36	C₇×#359	C₂²	1	2	yes	complemented
37	C₇×#350	C₂²	1	2	yes	complemented
38	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic, complemented
39	#33857	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₇×2₋¹⁺⁴	—	2	2	no	—
41	C₇×H(5,𝔽₂)	—	2	2	no	—
42	C₇×#137	—	2	2	no	—
43	C₇×#137	—	2	2	no	—
44	C₇×#137	—	4	2	no	—
45	C₇×#137	—	4	2	no	—
46	C₇×AGL(1,ℤ/8)	—	2	2	no	—
47	C₇×AGL(1,ℤ/8)	—	2	2	no	—
48	C₇×AGL(1,ℤ/8)	—	4	2	no	—
49	C₇×AGL(1,ℤ/8)	—	4	2	no	—
50	C₇×#135	—	2	2	no	—
51	C₇×#135	—	2	2	no	—
52	C₇×#135	—	4	2	no	—
53	C₇×#135	D₄	1	1	yes	characteristic
54	C₇×#135	C₂³	1	2	yes	—
55	C₂×C₇×Q₁₆	—	2	2	no	—
56	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
57	C₂×C₇×SD₁₆	—	2	2	no	—
58	C₂×C₇×SD₁₆	—	4	2	no	—
59	C₂×C₇×D₈	—	2	2	no	—
60	C₂×C₇×D₈	C₂³	1	2	yes	—
61	C₇×#131	—	2	2	no	—
62	C₇×#131	C₂³	1	2	yes	—
63	C₇×#124	—	2	2	no	—
64	C₇×Q₃₂	—	2	2	no	—
65	C₇×Q₃₂	—	4	1	no	—
66	C₇×Q₃₂	D₄	1	1	yes	characteristic
67	C₇×SD₃₂	—	2	2	no	—
68	C₇×SD₃₂	—	2	2	no	—
69	C₇×SD₃₂	—	2	2	no	—
70	C₇×SD₃₂	—	2	2	no	—
71	C₇×SD₃₂	D₄	1	2	yes	—
72	C₇×D₁₆	—	2	2	no	—
73	C₇×D₁₆	—	4	1	no	—
74	C₇×D₁₆	D₄	1	1	yes	characteristic
75	C₇×M₃₂	—	2	2	no	—
76	C₇×M₃₂	C₂³	1	2	yes	—
77	C₂×C₇×C₁₆	—	2	2	no	—
78	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
79	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
80	C₇×#108	D₄	1	1	yes	characteristic
81	C₇×#108	C₂³	1	2	yes	—
82	C₇×#104	—	2	2	no	—
83	C₇×#104	D₄	1	2	yes	—
84	C₇×#101	—	2	2	no	—
85	C₇×#100	—	2	2	no	—
86	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
87	#3416	C₂×C₇	1	2	yes	complemented
88	#3290	C₂×C₇	1	2	yes	complemented
89	#2249	C₂×C₇	1	1	yes	characteristic, complemented
90	#2223	C₂×C₇	1	2	yes	complemented
91	#2212	C₂×C₇	1	2	yes	complemented
92	#2195	C₂×C₇	1	2	yes	complemented
93	#2192	C₂×C₇	1	2	yes	complemented
94	#2178	C₂×C₇	1	1	yes	characteristic, complemented
95	#2170	C₂×C₇	1	1	yes	characteristic, complemented
96	C₇×#41	—	2	2	no	—
97	C₇×#41	—	4	2	no	—
98	C₇×#41	—	4	2	no	—
99	C₇×#41	—	4	2	no	—
100	C₇×#41	—	4	2	no	—
101	C₇×#41	—	8	2	no	—
102	C₇×#41	C₂×D₄	1	2	yes	—
103	C₂×C₇×Q₈	—	2	2	no	—
104	C₂×C₇×Q₈	—	4	2	no	—
105	C₂×C₇×Q₈	—	4	2	no	—
106	C₂×C₇×D₄	—	2	2	no	—
107	C₂×C₇×D₄	—	4	2	no	—
108	C₂×C₇×D₄	—	4	2	no	—
109	C₂×C₇×D₄	—	8	2	no	—
110	C₇×Q₁₆	—	2	2	no	—
111	C₇×Q₁₆	—	2	2	no	—
112	C₇×Q₁₆	—	4	2	no	—
113	C₇×Q₁₆	—	4	2	no	—
114	C₇×Q₁₆	D₈	1	1	yes	characteristic
115	C₇×Q₁₆	C₂×D₄	1	2	yes	—
116	C₇×SD₁₆	—	2	2	no	—
117	C₇×SD₁₆	—	4	2	no	—
118	C₇×SD₁₆	—	4	2	no	—
119	C₇×SD₁₆	—	4	2	no	—
120	C₇×SD₁₆	—	4	2	no	—
121	C₇×SD₁₆	—	4	2	no	—
122	C₇×SD₁₆	—	4	2	no	—
123	C₇×SD₁₆	D₈	1	2	yes	—
124	C₇×D₈	—	2	2	no	—
125	C₇×D₈	—	2	2	no	—
126	C₇×D₈	—	4	2	no	—
127	C₇×D₈	—	4	2	no	—
128	C₇×D₈	D₈	1	1	yes	characteristic
129	C₇×D₈	C₂×D₄	1	2	yes	—
130	C₇×M₁₆	—	2	2	no	—
131	C₇×M₁₆	—	2	2	no	—
132	C₇×M₁₆	—	4	2	no	—
133	C₇×M₁₆	C₂×D₄	1	2	yes	—
134	C₂×C₇×C₈	—	2	2	no	—
135	C₂×C₇×C₈	—	4	2	no	—
136	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
137	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
138	C₇×C₂²⋊C₄	—	8	2	no	—
139	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
140	C₇×C₁₆	—	2	2	no	—
141	C₇×C₁₆	—	2	1	no	—
142	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
143	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
144	#578	C₂²×C₇	1	2	yes	—
145	#577	—	2	2	no	—
146	#576	C₂²×C₇	1	2	yes	complemented
147	#510	—	2	2	no	—
148	#510	C₂²×C₇	1	2	yes	complemented
149	#509	—	2	2	no	—
150	#509	C₂²×C₇	1	2	yes	—
151	#508	—	2	2	no	—
152	#508	C₂²×C₇	1	1	yes	characteristic
153	C₂×Q₃₂	C₂²×C₇	1	1	yes	characteristic, complemented
154	C₂×SD₃₂	—	2	2	no	—
155	C₂×SD₃₂	C₂²×C₇	1	2	yes	complemented
156	C₂×D₁₆	C₂²×C₇	1	1	yes	characteristic
157	#504	C₂²×C₇	1	2	yes	complemented
158	#495	C₂²×C₇	1	2	yes	complemented
159	#473	C₂²×C₇	1	2	yes	complemented
160	#472	C₂²×C₇	1	2	yes	—
161	#471	—	2	2	no	—
162	#455	—	2	2	no	—
163	#454	—	2	2	no	—
164	#443	C₂²×C₇	1	1	yes	characteristic
165	#443	C₂²×C₇	1	2	yes	complemented
166	#368	C₂²×C₇	1	1	yes	characteristic, complemented
167	#364	C₂²×C₇	1	1	yes	characteristic, complemented
168	#362	C₂²×C₇	1	2	yes	complemented
169	#361	C₂²×C₇	1	2	yes	complemented
170	#359	C₂²×C₇	1	2	yes	complemented
171	#350	C₂²×C₇	1	2	yes	complemented
172	C₄×C₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
173	C₂³×C₇	—	4	2	no	—
174	C₂³×C₇	—	8	2	no	—
175	C₇×Q₈	—	2	2	no	—
176	C₇×Q₈	—	4	2	no	—
177	C₇×Q₈	—	4	2	no	—
178	C₇×Q₈	—	8	2	no	—
179	C₇×Q₈	C₂×D₈	1	2	yes	—
180	C₇×D₄	—	2	2	no	—
181	C₇×D₄	—	2	2	no	—
182	C₇×D₄	—	4	2	no	—
183	C₇×D₄	—	4	2	no	—
184	C₇×D₄	—	4	2	no	—
185	C₇×D₄	—	8	2	no	—
186	C₇×D₄	—	8	2	no	—
187	C₇×D₄	—	8	2	no	—
188	C₇×D₄	C₂×D₈	1	2	yes	—
189	C₂×C₄×C₇	—	2	2	no	—
190	C₂×C₄×C₇	—	2	1	no	—
191	C₂×C₄×C₇	—	4	2	no	—
192	C₂×C₄×C₇	—	4	2	no	—
193	C₂×C₄×C₇	—	8	2	no	—
194	C₂×C₄×C₇	—	8	2	no	—
195	C₂×C₄×C₇	—	8	2	no	—
196	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
197	C₇×C₈	—	2	2	no	—
198	C₇×C₈	—	4	2	no	—
199	C₇×C₈	C₂×D₈	1	1	yes	characteristic
200	C₇×C₈	C₂×D₈	1	1	yes	characteristic
201	C₇×C₈	C₂²×D₄	1	1	yes	characteristic
202	C₇×C₈	H(5,𝔽₂)	1	1	yes	characteristic
203	2₋¹⁺⁴	—	2	2	no	—
204	H(5,𝔽₂)	—	2	2	no	—
205	#137	—	2	2	no	—
206	#137	—	2	2	no	—
207	#137	—	4	2	no	—
208	#137	—	4	2	no	—
209	AGL(1,ℤ/8)	—	2	2	no	—
210	AGL(1,ℤ/8)	—	2	2	no	—
211	AGL(1,ℤ/8)	—	4	2	no	—
212	AGL(1,ℤ/8)	—	4	2	no	—
213	#135	—	2	2	no	—
214	#135	—	2	2	no	—
215	#135	—	4	2	no	—
216	#135	C₇×D₄	1	1	yes	characteristic
217	#135	C₂³×C₇	1	2	yes	—
218	C₂×Q₁₆	—	2	2	no	—
219	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
220	C₂×SD₁₆	—	2	2	no	—
221	C₂×SD₁₆	—	4	2	no	—
222	C₂×D₈	—	2	2	no	—
223	C₂×D₈	C₂³×C₇	1	2	yes	—
224	#131	—	2	2	no	—
225	#131	C₂³×C₇	1	2	yes	—
226	#124	—	2	2	no	—
227	Q₃₂	—	2	2	no	—
228	Q₃₂	—	4	1	no	—
229	Q₃₂	C₇×D₄	1	1	yes	characteristic
230	SD₃₂	—	2	2	no	—
231	SD₃₂	—	2	2	no	—
232	SD₃₂	—	2	2	no	—
233	SD₃₂	—	2	2	no	—
234	SD₃₂	C₇×D₄	1	2	yes	—
235	D₁₆	—	2	2	no	—
236	D₁₆	—	4	1	no	—
237	D₁₆	C₇×D₄	1	1	yes	characteristic
238	M₃₂	—	2	2	no	—
239	M₃₂	C₂³×C₇	1	2	yes	—
240	C₂×C₁₆	—	2	2	no	—
241	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
242	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
243	#108	C₇×D₄	1	1	yes	characteristic
244	#108	C₂³×C₇	1	2	yes	—
245	#104	—	2	2	no	—
246	#104	C₇×D₄	1	2	yes	—
247	#101	—	2	2	no	—
248	#100	—	2	2	no	—
249	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
250	C₂²×C₇	—	2	2	no	—
251	C₂²×C₇	—	4	2	no	—
252	C₂²×C₇	—	8	2	no	—
253	C₂²×C₇	—	16	2	no	—
254	C₂²×C₇	—	32	2	no	—
255	C₂²×C₇	#576	1	1	yes	characteristic
256	C₄×C₇	—	2	2	no	—
257	C₄×C₇	—	4	1	no	—
258	C₄×C₇	—	4	2	no	—
259	C₄×C₇	—	8	2	no	—
260	C₄×C₇	D₄²	1	1	yes	characteristic
261	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
262	#41	—	2	2	no	—
263	#41	—	4	2	no	—
264	#41	—	4	2	no	—
265	#41	—	4	2	no	—
266	#41	—	4	2	no	—
267	#41	—	8	2	no	—
268	#41	C₂×C₇×D₄	1	2	yes	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	4	2	no	—
271	C₂×Q₈	—	4	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	4	2	no	—
274	C₂×D₄	—	4	2	no	—
275	C₂×D₄	—	8	2	no	—
276	Q₁₆	—	2	2	no	—
277	Q₁₆	—	2	2	no	—
278	Q₁₆	—	4	2	no	—
279	Q₁₆	—	4	2	no	—
280	Q₁₆	C₇×D₈	1	1	yes	characteristic
281	Q₁₆	C₂×C₇×D₄	1	2	yes	—
282	SD₁₆	—	2	2	no	—
283	SD₁₆	—	4	2	no	—
284	SD₁₆	—	4	2	no	—
285	SD₁₆	—	4	2	no	—
286	SD₁₆	—	4	2	no	—
287	SD₁₆	—	4	2	no	—
288	SD₁₆	—	4	2	no	—
289	SD₁₆	C₇×D₈	1	2	yes	—
290	D₈	—	2	2	no	—
291	D₈	—	2	2	no	—
292	D₈	—	4	2	no	—
293	D₈	—	4	2	no	—
294	D₈	C₇×D₈	1	1	yes	characteristic
295	D₈	C₂×C₇×D₄	1	2	yes	—
296	M₁₆	—	2	2	no	—
297	M₁₆	—	2	2	no	—
298	M₁₆	—	4	2	no	—
299	M₁₆	C₂×C₇×D₄	1	2	yes	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	4	2	no	—
302	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
303	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
304	C₂²⋊C₄	—	8	2	no	—
305	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
306	C₁₆	—	2	2	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	C₂×C₇×D₄	1	1	yes	characteristic
309	C₁₆	C₂×C₇×D₄	1	1	yes	characteristic
310	C₂×C₇	—	2	1	no	—
311	C₂×C₇	—	4	2	no	—
312	C₂×C₇	—	8	2	no	—
313	C₂×C₇	—	16	2	no	—
314	C₂×C₇	D₄×D₈	1	1	yes	characteristic
315	C₂³	—	4	2	no	—
316	C₂³	—	8	2	no	—
317	Q₈	—	2	2	no	—
318	Q₈	—	4	2	no	—
319	Q₈	—	4	2	no	—
320	Q₈	—	8	2	no	—
321	Q₈	C₂×C₇×D₈	1	2	yes	—
322	D₄	—	2	2	no	—
323	D₄	—	2	2	no	—
324	D₄	—	4	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	8	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	8	2	no	—
330	D₄	C₂×C₇×D₈	1	2	yes	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	8	2	no	—
336	C₂×C₄	—	8	2	no	—
337	C₂×C₄	—	8	2	no	—
338	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	4	2	no	—
341	C₈	C₂×C₇×D₈	1	1	yes	characteristic
342	C₈	C₂×C₇×D₈	1	1	yes	characteristic
343	C₈	C₂²×C₇×D₄	1	1	yes	characteristic
344	C₈	C₇×H(5,𝔽₂)	1	1	yes	characteristic
345	C₇	#33857	1	1	yes	characteristic, direct product factor, Sylow
346	C₂²	—	2	2	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	16	2	no	—
350	C₂²	—	32	2	no	—
351	C₂²	C₇×#576	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	2	no	—
355	C₄	—	8	2	no	—
356	C₄	C₇×D₄²	1	1	yes	characteristic
357	C₄	C₂²×C₇×D₈	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	4	2	no	—
360	C₂	—	8	2	no	—
361	C₂	—	16	2	no	—
362	C₂	C₇×D₄×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#443 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3416	C₂	semidirect
2. C₇×#3290	C₂	semidirect
3. C₇×#2249	C₂	semidirect
4. C₇×#2223	C₂	semidirect
5. C₇×#2212	C₂	semidirect
6. C₇×#2195	C₂	semidirect
7. C₇×#2192	C₂	semidirect
8. C₇×#2178	C₂	semidirect
9. C₇×#2170	C₂	semidirect
12. C₇×#576	C₂²	semidirect
14. C₇×#510	C₂²	semidirect
19. C₂×C₇×Q₃₂	C₂²	semidirect
21. C₂×C₇×SD₃₂	C₂²	semidirect
23. C₇×#504	C₂²	semidirect
24. C₇×#495	C₂²	semidirect
25. C₇×#473	C₂²	semidirect
31. C₇×#443	C₂²	semidirect
32. C₇×#368	C₂²	semidirect
33. C₇×#364	C₂²	semidirect
34. C₇×#362	C₂²	semidirect
35. C₇×#361	C₂²	semidirect
36. C₇×#359	C₂²	semidirect
37. C₇×#350	C₂²	semidirect
38. C₄×C₇×C₁₆	C₂²	semidirect
39. #33857	C₇	direct (N × H)
87. #3416	C₂×C₇	semidirect
88. #3290	C₂×C₇	semidirect
89. #2249	C₂×C₇	semidirect
90. #2223	C₂×C₇	semidirect
91. #2212	C₂×C₇	semidirect
92. #2195	C₂×C₇	semidirect
93. #2192	C₂×C₇	semidirect
94. #2178	C₂×C₇	semidirect
95. #2170	C₂×C₇	semidirect
146. #576	C₂²×C₇	semidirect
148. #510	C₂²×C₇	semidirect
153. C₂×Q₃₂	C₂²×C₇	semidirect
155. C₂×SD₃₂	C₂²×C₇	semidirect
157. #504	C₂²×C₇	semidirect
158. #495	C₂²×C₇	semidirect
159. #473	C₂²×C₇	semidirect
165. #443	C₂²×C₇	semidirect
166. #368	C₂²×C₇	semidirect
167. #364	C₂²×C₇	semidirect
168. #362	C₂²×C₇	semidirect
169. #361	C₂²×C₇	semidirect
170. #359	C₂²×C₇	semidirect
171. #350	C₂²×C₇	semidirect
172. C₄×C₁₆	C₂²×C₇	semidirect
345. C₇	#33857	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×#3416	2
C₇×#3290	2
C₇×#2249	2
C₇×#2223	2
C₇×#2212	2
C₇×#2195	2
C₇×#2192	2
C₇×#2178	2
C₇×#2170	2
#33857	7

Subgroups — C7×#33857 #13211073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#33857 #13211073