Subgroups — C₇×#33866 #13211082

Order 1792.

548 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3338	C₂	1	1	yes	maximal, characteristic
2	C₇×D₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×#2241	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2236	C₂	1	2	yes	maximal, complemented
5	C₇×#2227	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2216	C₂	1	2	yes	maximal, complemented
7	C₇×#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2188	C₂	1	2	yes	maximal, complemented
9	C₄×C₇×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
11	C₂×C₇×#358	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
13	C₇×#554	—	2	1	no	—
14	C₇×D₄×Q₈	—	2	1	no	—
15	C₂×C₇×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₇×Q₃₂	C₂²	1	2	yes	—
17	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
18	C₇×#500	C₂²	1	1	yes	characteristic
19	C₇×#494	C₂²	1	1	yes	characteristic
20	C₇×#484	—	2	2	no	—
21	C₇×#477	—	2	1	no	—
22	C₇×#467	C₂²	1	2	yes	—
23	C₇×#467	C₂²	1	2	yes	—
24	C₇×#462	—	2	1	no	—
25	C₇×#451	—	2	2	no	—
26	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
28	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
29	C₂×C₇×#107	C₂²	1	2	yes	—
30	C₇×#366	C₂²	1	2	yes	—
31	C₇×#366	C₂²	1	1	yes	characteristic
32	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₇×#358	C₂²	1	2	yes	—
34	C₇×#358	C₂²	1	2	yes	—
35	C₇×#358	C₂²	1	2	yes	—
36	C₇×#358	C₂²	1	2	yes	—
37	C₇×#358	C₂²	1	1	yes	characteristic
38	C₇×#358	C₂²	1	1	yes	characteristic
39	C₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
41	#33866	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂²×C₇×Q₈	—	2	2	no	—
43	C₂×C₇×Q₁₆	—	2	4	no	—
44	C₂×C₇×Q₁₆	—	2	2	no	—
45	C₂×C₇×Q₁₆	—	2	1	no	—
46	C₂×C₇×Q₁₆	—	4	2	no	—
47	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
48	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
49	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
50	C₂²×C₇×C₈	C₂³	1	2	yes	—
51	C₇×C₄⋊Q₈	—	2	1	no	—
52	C₇×C₄⋊Q₈	—	2	1	no	—
53	C₇×C₄⋊Q₈	—	4	1	no	—
54	C₇×#125	—	4	1	no	—
55	C₇×C₂²⋊Q₈	—	2	2	no	—
56	C₇×C₂²⋊Q₈	—	2	2	no	—
57	C₇×C₂²⋊Q₈	—	4	2	no	—
58	C₇×C₂²⋊Q₈	—	4	2	no	—
59	C₄×C₇×Q₈	—	2	1	no	—
60	C₄×C₇×Q₈	—	2	1	no	—
61	C₄×C₇×D₄	—	4	1	no	—
62	C₄×C₇×D₄	—	4	1	no	—
63	C₄×C₇×D₄	D₄	1	1	yes	characteristic
64	C₂×C₇×C₄⋊C₄	—	2	2	no	—
65	C₇×Q₃₂	—	2	2	no	—
66	C₇×Q₃₂	—	4	2	no	—
67	C₂×C₇×C₁₆	—	4	2	no	—
68	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₇×C₁₆	C₂³	1	2	yes	—
70	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
71	C₇×#107	—	2	2	no	—
72	C₇×#107	—	2	1	no	—
73	C₇×#107	C₂³	1	1	yes	characteristic
74	C₇×#107	C₂³	1	2	yes	—
75	C₇×#107	C₂³	1	1	yes	characteristic
76	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
77	C₇×#103	—	2	2	no	—
78	C₇×#103	—	2	2	no	—
79	C₇×#103	—	2	1	no	—
80	C₇×#103	—	2	1	no	—
81	C₇×C₂²⋊C₈	D₄	1	2	yes	—
82	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
83	#3338	C₂×C₇	1	1	yes	characteristic
84	D₄×Q₁₆	C₂×C₇	1	1	yes	characteristic
85	#2241	C₂×C₇	1	1	yes	characteristic, complemented
86	#2236	C₂×C₇	1	2	yes	complemented
87	#2227	C₂×C₇	1	1	yes	characteristic, complemented
88	#2216	C₂×C₇	1	2	yes	complemented
89	#2209	C₂×C₇	1	1	yes	characteristic, complemented
90	#2188	C₂×C₇	1	2	yes	complemented
91	C₄×Q₃₂	C₂×C₇	1	1	yes	characteristic, complemented
92	D₄×C₁₆	C₂×C₇	1	1	yes	characteristic
93	C₂×#358	C₂×C₇	1	2	yes	complemented
94	C₂×C₇×Q₈	—	2	1	no	—
95	C₂×C₇×Q₈	—	2	1	no	—
96	C₂×C₇×Q₈	—	2	2	no	—
97	C₂×C₇×Q₈	—	4	4	no	—
98	C₂×C₇×Q₈	—	4	2	no	—
99	C₂×C₇×Q₈	—	4	1	no	—
100	C₂×C₇×Q₈	—	8	2	no	—
101	C₂×C₇×D₄	D₈	1	1	yes	characteristic
102	C₂²×C₄×C₇	—	4	2	no	—
103	C₂²×C₄×C₇	—	4	2	no	—
104	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
105	C₇×Q₁₆	—	2	1	no	—
106	C₇×Q₁₆	—	2	2	no	—
107	C₇×Q₁₆	—	2	1	no	—
108	C₇×Q₁₆	—	2	1	no	—
109	C₇×Q₁₆	—	2	2	no	—
110	C₇×Q₁₆	—	4	4	no	—
111	C₇×Q₁₆	—	8	1	no	—
112	C₇×Q₁₆	#41	1	2	yes	—
113	C₂×C₇×C₈	—	2	2	no	—
114	C₂×C₇×C₈	—	2	2	no	—
115	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
117	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
118	C₇×C₄⋊C₄	—	2	1	no	—
119	C₇×C₄⋊C₄	—	2	1	no	—
120	C₇×C₄⋊C₄	—	2	2	no	—
121	C₇×C₄⋊C₄	—	4	2	no	—
122	C₇×C₄⋊C₄	—	4	1	no	—
123	C₇×C₄⋊C₄	—	4	1	no	—
124	C₇×C₄⋊C₄	—	4	1	no	—
125	C₇×C₄⋊C₄	—	4	1	no	—
126	C₇×C₄⋊C₄	—	4	2	no	—
127	C₇×C₄⋊C₄	—	4	2	no	—
128	C₇×C₄⋊C₄	—	4	1	no	—
129	C₇×C₄⋊C₄	D₈	1	1	yes	characteristic
130	C₇×C₂²⋊C₄	—	4	2	no	—
131	C₇×C₂²⋊C₄	—	4	2	no	—
132	C₇×C₂²⋊C₄	D₈	1	2	yes	—
133	C₄²×C₇	—	4	1	no	—
134	C₄²×C₇	—	4	1	no	—
135	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
136	C₇×C₁₆	—	2	1	no	—
137	C₇×C₁₆	—	2	1	no	—
138	C₇×C₁₆	—	2	2	no	—
139	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
140	#554	—	2	1	no	—
141	D₄×Q₈	—	2	1	no	—
142	C₂×Q₃₂	C₂²×C₇	1	1	yes	characteristic
143	C₂×Q₃₂	C₂²×C₇	1	2	yes	—
144	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
145	#500	C₂²×C₇	1	1	yes	characteristic
146	#494	C₂²×C₇	1	1	yes	characteristic
147	#484	—	2	2	no	—
148	#477	—	2	1	no	—
149	#467	C₂²×C₇	1	2	yes	—
150	#467	C₂²×C₇	1	2	yes	—
151	#462	—	2	1	no	—
152	#451	—	2	2	no	—
153	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
154	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
155	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
156	C₂×#107	C₂²×C₇	1	2	yes	—
157	#366	C₂²×C₇	1	1	yes	characteristic
158	#366	C₂²×C₇	1	2	yes	—
159	C₄⋊C₁₆	C₂²×C₇	1	1	yes	characteristic
160	#358	C₂²×C₇	1	1	yes	characteristic
161	#358	C₂²×C₇	1	2	yes	—
162	#358	C₂²×C₇	1	2	yes	—
163	#358	C₂²×C₇	1	2	yes	—
164	#358	C₂²×C₇	1	1	yes	characteristic
165	#358	C₂²×C₇	1	2	yes	—
166	C₂²⋊C₁₆	C₂²×C₇	1	2	yes	—
167	C₄×C₁₆	C₂²×C₇	1	1	yes	characteristic
168	C₂³×C₇	C₂×D₈	1	2	yes	—
169	C₇×Q₈	—	2	2	no	—
170	C₇×Q₈	—	4	1	no	—
171	C₇×Q₈	—	4	2	no	—
172	C₇×Q₈	—	4	1	no	—
173	C₇×Q₈	—	4	2	no	—
174	C₇×Q₈	—	4	1	no	—
175	C₇×Q₈	—	8	4	no	—
176	C₇×Q₈	—	16	1	no	—
177	C₇×D₄	Q₃₂	1	4	yes	complemented
178	C₂×C₄×C₇	—	2	2	no	—
179	C₂×C₄×C₇	—	2	2	no	—
180	C₂×C₄×C₇	—	4	1	no	—
181	C₂×C₄×C₇	—	4	1	no	—
182	C₂×C₄×C₇	—	4	1	no	—
183	C₂×C₄×C₇	—	4	1	no	—
184	C₂×C₄×C₇	—	4	2	no	—
185	C₂×C₄×C₇	—	4	2	no	—
186	C₂×C₄×C₇	—	8	4	no	—
187	C₂×C₄×C₇	—	16	2	no	—
188	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
189	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
190	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
191	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
192	C₇×C₈	—	2	2	no	—
193	C₇×C₈	—	2	1	no	—
194	C₇×C₈	C₂×#41	1	1	yes	characteristic
195	C₇×C₈	H(5,𝔽₂)	1	1	yes	characteristic
196	C₂²×Q₈	—	2	2	no	—
197	C₂×Q₁₆	—	2	4	no	—
198	C₂×Q₁₆	—	2	2	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	—	4	2	no	—
201	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
202	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
203	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₇	1	2	yes	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	4	1	no	—
208	#125	—	4	1	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₂²⋊Q₈	—	4	2	no	—
212	C₂²⋊Q₈	—	4	2	no	—
213	C₄×Q₈	—	2	1	no	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×D₄	—	4	1	no	—
216	C₄×D₄	—	4	1	no	—
217	C₄×D₄	C₇×D₄	1	1	yes	characteristic
218	C₂×C₄⋊C₄	—	2	2	no	—
219	Q₃₂	—	2	2	no	—
220	Q₃₂	—	4	2	no	—
221	C₂×C₁₆	—	4	2	no	—
222	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
223	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
224	C₂×C₁₆	C₂³×C₇	1	2	yes	—
225	#107	—	2	2	no	—
226	#107	—	2	1	no	—
227	#107	C₂³×C₇	1	1	yes	characteristic
228	#107	C₂³×C₇	1	2	yes	—
229	#107	C₂³×C₇	1	1	yes	characteristic
230	C₄⋊C₈	C₇×D₄	1	1	yes	characteristic
231	#103	—	2	1	no	—
232	#103	—	2	1	no	—
233	#103	—	2	2	no	—
234	#103	—	2	2	no	—
235	C₂²⋊C₈	C₇×D₄	1	2	yes	—
236	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
237	C₂²×C₇	—	2	2	no	—
238	C₂²×C₇	—	2	2	no	—
239	C₂²×C₇	C₂×Q₃₂	1	4	yes	—
240	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
241	C₄×C₇	—	2	1	no	—
242	C₄×C₇	—	2	2	no	—
243	C₄×C₇	—	4	2	no	—
244	C₄×C₇	—	8	1	no	—
245	C₄×C₇	—	8	1	no	—
246	C₄×C₇	—	8	2	no	—
247	C₄×C₇	—	8	2	no	—
248	C₄×C₇	—	8	1	no	—
249	C₄×C₇	C₂×Q₃₂	1	2	yes	—
250	C₄×C₇	#546	1	1	yes	characteristic
251	C₄×C₇	#577	1	1	yes	characteristic
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	4	4	no	—
256	C₂×Q₈	—	4	2	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	8	2	no	—
259	C₂×D₄	C₇×D₈	1	1	yes	characteristic
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
263	Q₁₆	—	2	1	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	2	1	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	4	4	no	—
269	Q₁₆	—	8	1	no	—
270	Q₁₆	C₇×#41	1	2	yes	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
274	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
275	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	C₇×D₈	1	1	yes	characteristic
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	C₇×D₈	1	2	yes	—
291	C₄²	—	4	1	no	—
292	C₄²	—	4	1	no	—
293	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
294	C₁₆	—	2	1	no	—
295	C₁₆	—	2	1	no	—
296	C₁₆	—	2	2	no	—
297	C₂×C₇	—	2	4	no	—
298	C₂×C₇	#3294	1	1	yes	characteristic
299	C₂×C₇	C₂²×Q₃₂	1	1	yes	characteristic
300	C₂×C₇	#3416	1	1	yes	characteristic
301	C₂³	C₂×C₇×D₈	1	2	yes	—
302	Q₈	—	2	2	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	4	1	no	—
308	Q₈	—	8	4	no	—
309	Q₈	—	16	1	no	—
310	D₄	C₇×Q₃₂	1	4	yes	complemented
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	—	16	2	no	—
321	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
322	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
323	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	1	no	—
327	C₈	C₂×C₇×#41	1	1	yes	characteristic
328	C₈	C₇×H(5,𝔽₂)	1	1	yes	characteristic
329	C₇	#33866	1	1	yes	characteristic, direct product factor, Sylow
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	C₂×C₇×Q₃₂	1	4	yes	—
333	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	2	no	—
336	C₄	—	4	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	2	no	—
339	C₄	—	8	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	1	no	—
342	C₄	C₂×C₇×Q₃₂	1	2	yes	—
343	C₄	C₇×#546	1	1	yes	characteristic
344	C₄	C₇×#577	1	1	yes	characteristic
345	C₂	—	2	4	no	—
346	C₂	C₇×#3294	1	1	yes	characteristic
347	C₂	C₂²×C₇×Q₃₂	1	1	yes	characteristic
348	C₂	C₇×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
prime radical	C₂×C₇×D₄	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×D₄ ◁ C₇×C₈×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#2241	C₂	semidirect
4. C₇×#2236	C₂	semidirect
5. C₇×#2227	C₂	semidirect
6. C₇×#2216	C₂	semidirect
7. C₇×#2209	C₂	semidirect
8. C₇×#2188	C₂	semidirect
9. C₄×C₇×Q₃₂	C₂	semidirect
11. C₂×C₇×#358	C₂	semidirect
41. #33866	C₇	direct (N × H)
85. #2241	C₂×C₇	semidirect
86. #2236	C₂×C₇	semidirect
87. #2227	C₂×C₇	semidirect
88. #2216	C₂×C₇	semidirect
89. #2209	C₂×C₇	semidirect
90. #2188	C₂×C₇	semidirect
91. C₄×Q₃₂	C₂×C₇	semidirect
93. C₂×#358	C₂×C₇	semidirect
177. C₇×D₄	Q₃₂	semidirect
310. D₄	C₇×Q₃₂	semidirect
329. C₇	#33866	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×#3338	2
C₇×D₄×Q₁₆	2
C₇×#2241	2
C₇×#2236	2
C₇×#2227	2
C₇×#2216	2
C₇×#2209	2
C₇×#2188	2
C₄×C₇×Q₃₂	2
C₇×D₄×C₁₆	2
C₂×C₇×#358	2
#33866	7

Subgroups — C7×#33866 #13211082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#33866 #13211082