Subgroups — C₇×#21297 #13198513

Order 1792.

738 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2943	C₂	1	1	yes	maximal, characteristic
2	C₇×#2923	C₂	1	2	yes	maximal, complemented
3	C₇×#1975	C₂	1	1	yes	maximal, characteristic
4	C₇×#1974	C₂	1	1	yes	maximal, characteristic
5	C₇×#1946	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1945	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1938	C₂	1	2	yes	maximal, complemented
8	C₇×#1937	C₂	1	2	yes	maximal, complemented
9	C₇×#1778	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₇×#104	C₂	1	2	yes	maximal, complemented
11	C₄×C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#135	C₂²	1	1	yes	characteristic
13	C₄×C₇×#41	C₂²	1	2	yes	—
14	C₇×#442	C₄	1	2	yes	complemented
15	C₇×#441	C₄	1	2	yes	complemented
16	C₇×#440	C₄	1	4	yes	complemented
17	C₇×#433	—	2	4	no	—
18	C₇×#432	—	2	4	no	—
19	C₂×C₇×#108	C₂²	1	1	yes	characteristic
20	C₂×C₇×#108	C₂²	1	1	yes	characteristic
21	C₇×#427	C₂²	1	1	yes	characteristic
22	C₇×#424	C₂²	1	2	yes	—
23	C₇×#424	C₂²	1	2	yes	—
24	C₂×C₇×#104	C₂²	1	2	yes	—
25	C₂×C₇×#104	C₂²	1	2	yes	—
26	C₇×#416	C₂²	1	2	yes	—
27	C₇×#408	C₂²	1	2	yes	—
28	C₇×#408	C₂²	1	2	yes	—
29	C₄×C₇×M₁₆	C₂²	1	2	yes	—
30	C₂×C₇×#97	C₂²	1	1	yes	characteristic
31	C₂×C₇×#97	C₂²	1	1	yes	characteristic
32	C₂×C₇×#97	C₂²	1	1	yes	characteristic
33	C₂×C₇×#97	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
36	C₄³×C₇	C₂²	1	1	yes	characteristic
37	C₇×#341	C₂²	1	2	yes	—
38	C₇×#339	C₂²	1	2	yes	—
39	C₇×#339	C₂²	1	2	yes	—
40	#21297	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂×C₇×#41	C₂³	1	2	yes	—
42	C₇×#135	—	2	2	no	—
43	C₇×#135	—	2	2	no	—
44	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
46	C₂×C₇×D₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₇×M₁₆	C₂³	1	2	yes	—
48	C₂×C₇×M₁₆	C₂³	1	2	yes	—
49	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
54	C₄×C₇×D₄	—	2	4	no	—
55	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
56	C₇×#117	—	2	2	no	—
57	C₇×#117	C₂³	1	2	yes	—
58	C₂×C₄²×C₇	—	2	2	no	—
59	C₂×C₄²×C₇	—	2	2	no	—
60	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
63	C₇×#108	—	2	2	no	—
64	C₇×#108	—	2	2	no	—
65	C₇×#107	C₂×C₄	1	2	yes	—
66	C₇×#106	C₂×C₄	1	2	yes	—
67	C₇×C₄⋊C₈	—	2	4	no	—
68	C₇×C₄⋊C₈	—	2	4	no	—
69	C₇×#104	—	2	4	no	—
70	C₇×#104	—	2	4	no	—
71	C₇×#103	C₂×C₄	1	4	yes	—
72	C₇×#102	C₂×C₄	1	4	yes	—
73	C₇×C₂²⋊C₈	—	2	4	no	—
74	C₇×C₂²⋊C₈	—	2	4	no	—
75	C₇×#97	—	2	2	no	—
76	C₇×#97	—	2	2	no	—
77	C₇×#97	—	2	2	no	—
78	C₇×#97	C₂×C₄	1	2	yes	—
79	C₇×#97	C₂×C₄	1	2	yes	—
80	C₇×#97	D₄	1	4	yes	—
81	C₄×C₇×C₈	—	2	2	no	—
82	C₄×C₇×C₈	—	2	2	no	—
83	C₄×C₇×C₈	—	2	2	no	—
84	#2943	C₂×C₇	1	1	yes	characteristic
85	#2923	C₂×C₇	1	2	yes	complemented
86	#1975	C₂×C₇	1	1	yes	characteristic
87	#1974	C₂×C₇	1	1	yes	characteristic
88	#1946	C₂×C₇	1	1	yes	characteristic, complemented
89	#1945	C₂×C₇	1	1	yes	characteristic, complemented
90	#1938	C₂×C₇	1	2	yes	complemented
91	#1937	C₂×C₇	1	2	yes	complemented
92	#1778	C₂×C₇	1	1	yes	characteristic, complemented
93	C₄×#104	C₂×C₇	1	2	yes	complemented
94	C₄×#97	C₂×C₇	1	1	yes	characteristic, complemented
95	C₇×#41	—	2	2	no	—
96	C₇×#41	—	2	2	no	—
97	C₇×#41	—	2	2	no	—
98	C₇×#41	—	2	2	no	—
99	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
100	C₂×C₇×D₄	—	2	2	no	—
101	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₇	—	2	1	no	—
103	C₂²×C₄×C₇	—	2	1	no	—
104	C₂²×C₄×C₇	—	2	2	no	—
105	C₂²×C₄×C₇	—	2	4	no	—
106	C₂²×C₄×C₇	#41	1	2	yes	—
107	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
108	C₇×Q₁₆	—	2	2	no	—
109	C₇×SD₁₆	—	2	4	no	—
110	C₇×D₈	—	2	2	no	—
111	C₇×M₁₆	—	2	2	no	—
112	C₇×M₁₆	—	2	2	no	—
113	C₇×M₁₆	—	2	2	no	—
114	C₇×M₁₆	—	2	2	no	—
115	C₂×C₇×C₈	—	2	1	no	—
116	C₂×C₇×C₈	—	2	1	no	—
117	C₂×C₇×C₈	—	2	1	no	—
118	C₂×C₇×C₈	—	2	1	no	—
119	C₂×C₇×C₈	—	2	1	no	—
120	C₂×C₇×C₈	—	2	1	no	—
121	C₂×C₇×C₈	—	2	2	no	—
122	C₂×C₇×C₈	—	2	2	no	—
123	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
124	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
127	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
128	C₂×C₇×C₈	#41	1	2	yes	—
129	C₂×C₇×C₈	#41	1	2	yes	—
130	C₇×C₄⋊C₄	—	2	4	no	—
131	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
132	C₇×C₂²⋊C₄	—	2	2	no	—
133	C₇×C₂²⋊C₄	—	2	4	no	—
134	C₄²×C₇	—	2	4	no	—
135	C₄²×C₇	—	2	2	no	—
136	C₄²×C₇	—	2	1	no	—
137	C₄²×C₇	—	2	1	no	—
138	C₄²×C₇	—	4	4	no	—
139	C₄²×C₇	C₂²×C₄	1	2	yes	—
140	C₄²×C₇	C₂×D₄	1	2	yes	—
141	C₄²×C₇	#41	1	2	yes	—
142	C₄²×C₇	#41	1	2	yes	—
143	C₂×#135	C₂²×C₇	1	1	yes	characteristic
144	C₄×#41	C₂²×C₇	1	2	yes	—
145	#442	C₄×C₇	1	2	yes	complemented
146	#441	C₄×C₇	1	2	yes	complemented
147	#440	C₄×C₇	1	4	yes	complemented
148	#433	—	2	4	no	—
149	#432	—	2	4	no	—
150	C₂×#108	C₂²×C₇	1	1	yes	characteristic
151	C₂×#108	C₂²×C₇	1	1	yes	characteristic
152	#427	C₂²×C₇	1	1	yes	characteristic
153	#424	C₂²×C₇	1	2	yes	—
154	#424	C₂²×C₇	1	2	yes	—
155	C₂×#104	C₂²×C₇	1	2	yes	—
156	C₂×#104	C₂²×C₇	1	2	yes	—
157	#416	C₂²×C₇	1	2	yes	—
158	#408	C₂²×C₇	1	2	yes	—
159	#408	C₂²×C₇	1	2	yes	—
160	C₄×M₁₆	C₂²×C₇	1	2	yes	—
161	C₂×#97	C₂²×C₇	1	1	yes	characteristic
162	C₂×#97	C₂²×C₇	1	1	yes	characteristic
163	C₂×#97	C₂²×C₇	1	1	yes	characteristic
164	C₂×#97	C₂²×C₇	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
167	C₄³	C₂²×C₇	1	1	yes	characteristic
168	#341	C₂²×C₇	1	2	yes	—
169	#339	C₂²×C₇	1	2	yes	—
170	#339	C₂²×C₇	1	2	yes	—
171	C₂³×C₇	—	2	2	no	—
172	C₂³×C₇	C₂×#41	1	1	yes	characteristic
173	C₇×Q₈	—	2	2	no	—
174	C₇×Q₈	—	2	2	no	—
175	C₇×D₄	—	2	2	no	—
176	C₇×D₄	—	2	2	no	—
177	C₇×D₄	—	4	2	no	—
178	C₇×D₄	—	4	2	no	—
179	C₂×C₄×C₇	—	2	4	no	—
180	C₂×C₄×C₇	—	2	2	no	—
181	C₂×C₄×C₇	—	2	4	no	—
182	C₂×C₄×C₇	—	2	2	no	—
183	C₂×C₄×C₇	—	2	2	no	—
184	C₂×C₄×C₇	—	4	2	no	—
185	C₂×C₄×C₇	—	4	2	no	—
186	C₂×C₄×C₇	—	4	2	no	—
187	C₂×C₄×C₇	—	4	2	no	—
188	C₂×C₄×C₇	—	8	4	no	—
189	C₂×C₄×C₇	C₄×D₄	1	4	yes	—
190	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
191	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
196	C₇×C₈	—	2	2	no	—
197	C₇×C₈	—	2	2	no	—
198	C₇×C₈	—	2	2	no	—
199	C₇×C₈	—	4	2	no	—
200	C₇×C₈	—	4	2	no	—
201	C₇×C₈	#117	1	2	yes	—
202	C₇×C₈	#117	1	2	yes	—
203	C₂×#41	C₂³×C₇	1	2	yes	—
204	#135	—	2	2	no	—
205	#135	—	2	2	no	—
206	C₂×Q₁₆	C₂×C₄×C₇	1	1	yes	characteristic
207	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
208	C₂×D₈	C₂×C₄×C₇	1	1	yes	characteristic
209	C₂×M₁₆	C₂³×C₇	1	2	yes	—
210	C₂×M₁₆	C₂³×C₇	1	2	yes	—
211	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
215	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
216	C₄×D₄	—	2	4	no	—
217	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
218	#117	—	2	2	no	—
219	#117	C₂³×C₇	1	2	yes	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
224	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
225	#108	—	2	2	no	—
226	#108	—	2	2	no	—
227	#107	C₂×C₄×C₇	1	2	yes	—
228	#106	C₂×C₄×C₇	1	2	yes	—
229	C₄⋊C₈	—	2	4	no	—
230	C₄⋊C₈	—	2	4	no	—
231	#104	—	2	4	no	—
232	#104	—	2	4	no	—
233	#103	C₂×C₄×C₇	1	4	yes	—
234	#102	C₂×C₄×C₇	1	4	yes	—
235	C₂²⋊C₈	—	2	4	no	—
236	C₂²⋊C₈	—	2	4	no	—
237	#97	—	2	2	no	—
238	#97	—	2	2	no	—
239	#97	—	2	2	no	—
240	#97	C₂×C₄×C₇	1	2	yes	—
241	#97	C₂×C₄×C₇	1	2	yes	—
242	#97	C₇×D₄	1	4	yes	—
243	C₄×C₈	—	2	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	—	2	2	no	—
246	C₂²×C₇	—	2	2	no	—
247	C₂²×C₇	—	4	4	no	—
248	C₂²×C₇	—	4	2	no	—
249	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
250	C₂²×C₇	#529	1	1	yes	characteristic
251	C₂²×C₇	#529	1	1	yes	characteristic
252	C₄×C₇	—	2	4	no	—
253	C₄×C₇	—	2	4	no	—
254	C₄×C₇	—	4	2	no	—
255	C₄×C₇	—	4	4	no	—
256	C₄×C₇	—	4	2	no	—
257	C₄×C₇	C₂×#117	1	1	yes	characteristic
258	C₄×C₇	C₄×#41	1	1	yes	characteristic
259	C₄×C₇	#532	1	1	yes	characteristic
260	C₄×C₇	#532	1	1	yes	characteristic
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	2	2	no	—
264	#41	—	2	2	no	—
265	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	C₇×#41	1	2	yes	—
273	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
274	Q₁₆	—	2	2	no	—
275	SD₁₆	—	2	4	no	—
276	D₈	—	2	2	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
290	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
291	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
292	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
293	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
294	C₂×C₈	C₇×#41	1	2	yes	—
295	C₂×C₈	C₇×#41	1	2	yes	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	2	4	no	—
300	C₄²	—	2	4	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	1	no	—
304	C₄²	—	4	4	no	—
305	C₄²	C₂²×C₄×C₇	1	2	yes	—
306	C₄²	C₂×C₇×D₄	1	2	yes	—
307	C₄²	C₇×#41	1	2	yes	—
308	C₄²	C₇×#41	1	2	yes	—
309	C₂×C₇	—	2	1	no	—
310	C₂×C₇	—	2	1	no	—
311	C₂×C₇	—	8	2	no	—
312	C₂×C₇	#2293	1	1	yes	characteristic
313	C₂×C₇	#2956	1	2	yes	—
314	C₂³	—	2	2	no	—
315	C₂³	C₂×C₇×#41	1	1	yes	characteristic
316	Q₈	—	2	2	no	—
317	Q₈	—	2	2	no	—
318	D₄	—	2	2	no	—
319	D₄	—	2	2	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	2	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	8	4	no	—
332	C₂×C₄	C₄×C₇×D₄	1	4	yes	—
333	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
335	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
336	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
337	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
338	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	2	2	no	—
342	C₈	—	4	2	no	—
343	C₈	—	4	2	no	—
344	C₈	C₇×#117	1	2	yes	—
345	C₈	C₇×#117	1	2	yes	—
346	C₇	#21297	1	1	yes	characteristic, direct product factor, Sylow
347	C₂²	—	2	2	no	—
348	C₂²	—	4	4	no	—
349	C₂²	—	4	2	no	—
350	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
351	C₂²	C₇×#529	1	1	yes	characteristic
352	C₂²	C₇×#529	1	1	yes	characteristic
353	C₄	—	2	4	no	—
354	C₄	—	2	4	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	4	no	—
357	C₄	—	4	2	no	—
358	C₄	C₂×C₇×#117	1	1	yes	characteristic
359	C₄	C₄×C₇×#41	1	1	yes	characteristic
360	C₄	C₇×#532	1	1	yes	characteristic
361	C₄	C₇×#532	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	8	2	no	—
365	C₂	C₇×#2293	1	1	yes	characteristic
366	C₂	C₇×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂×C₇×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2923	C₂	semidirect
5. C₇×#1946	C₂	semidirect
6. C₇×#1945	C₂	semidirect
7. C₇×#1938	C₂	semidirect
8. C₇×#1937	C₂	semidirect
9. C₇×#1778	C₂	semidirect
10. C₄×C₇×#104	C₂	semidirect
11. C₄×C₇×#97	C₂	semidirect
14. C₇×#442	C₄	semidirect
15. C₇×#441	C₄	semidirect
16. C₇×#440	C₄	semidirect
40. #21297	C₇	direct (N × H)
85. #2923	C₂×C₇	semidirect
88. #1946	C₂×C₇	semidirect
89. #1945	C₂×C₇	semidirect
90. #1938	C₂×C₇	semidirect
91. #1937	C₂×C₇	semidirect
92. #1778	C₂×C₇	semidirect
93. C₄×#104	C₂×C₇	semidirect
94. C₄×#97	C₂×C₇	semidirect
145. #442	C₄×C₇	semidirect
146. #441	C₄×C₇	semidirect
147. #440	C₄×C₇	semidirect
346. C₇	#21297	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×#2943	2
C₇×#2923	2
C₇×#1975	2
C₇×#1974	2
C₇×#1946	2
C₇×#1945	2
C₇×#1938	2
C₇×#1937	2
C₇×#1778	2
C₄×C₇×#104	2
C₄×C₇×#97	2
#21297	7

Subgroups — C7×#21297 #13198513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21297 #13198513