Subgroups — C₇×#60388 #13237604

Order 1792.

4874 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×H(7,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3585	C₂	1	2	yes	maximal, complemented
3	C₇×#3498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3021	C₂	1	6	yes	maximal, complemented
5	C₇×#3019	C₂	1	6	yes	maximal, complemented
6	C₇×#3018	C₂	1	6	yes	maximal, complemented
7	C₂×C₇×#453	C₂	1	6	yes	maximal, complemented
8	C₇×#2901	C₂	1	2	yes	maximal, complemented
9	C₇×#2888	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#585	—	2	8	no	—
11	C₇×#585	—	2	2	no	—
12	C₇×#585	C₂²	1	2	yes	—
13	C₇×#585	C₂²	1	6	yes	complemented
14	C₂×C₇×H(5,𝔽₂)	—	2	8	no	—
15	C₂×C₇×H(5,𝔽₂)	—	2	6	no	—
16	C₂×C₇×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
17	C₂×C₇×H(5,𝔽₂)	C₂²	1	6	yes	complemented
18	C₂×C₇×H(5,𝔽₂)	C₂²	1	1	yes	characteristic, complemented
19	C₇×#577	—	2	8	no	—
20	C₇×#576	—	2	8	no	—
21	C₇×#575	C₂²	1	12	yes	complemented
22	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	12	yes	complemented
23	C₇×#568	C₂²	1	2	yes	complemented
24	C₇×#550	—	2	2	no	—
25	C₇×D₄²	—	2	6	no	—
26	C₇×#535	C₂²	1	3	yes	complemented
27	C₇×#535	C₂²	1	3	yes	complemented
28	C₇×#532	C₂²	1	3	yes	complemented
29	C₂×C₇×#127	C₂²	1	3	yes	complemented
30	C₄×C₇×#41	C₂²	1	1	yes	characteristic, complemented
31	C₇×#456	C₂²	1	4	yes	complemented
32	C₇×#455	C₂²	1	12	yes	complemented
33	C₇×#454	C₂²	1	12	yes	complemented
34	C₇×#453	C₂²	1	12	yes	complemented
35	C₇×#453	C₂²	1	24	yes	complemented
36	C₇×#421	C₂²	1	12	yes	complemented
37	C₂×C₇×#104	C₂²	1	12	yes	complemented
38	C₇×#413	C₂²	1	6	yes	complemented
39	C₂×C₇×#100	C₂²	1	6	yes	complemented
40	#60388	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₇×2₋¹⁺⁴	—	2	8	no	—
42	C₇×2₋¹⁺⁴	—	2	2	no	—
43	C₇×2₋¹⁺⁴	—	2	12	no	—
44	C₇×2₋¹⁺⁴	D₄	1	8	yes	complemented
45	C₇×2₋¹⁺⁴	C₂³	1	4	yes	—
46	C₇×H(5,𝔽₂)	—	2	8	no	—
47	C₇×H(5,𝔽₂)	—	2	48	no	—
48	C₇×H(5,𝔽₂)	—	2	8	no	—
49	C₇×H(5,𝔽₂)	—	2	24	no	—
50	C₇×H(5,𝔽₂)	—	2	6	no	—
51	C₇×H(5,𝔽₂)	—	2	12	no	—
52	C₇×H(5,𝔽₂)	—	2	12	no	—
53	C₇×H(5,𝔽₂)	—	2	12	no	—
54	C₇×H(5,𝔽₂)	—	2	8	no	—
55	C₇×H(5,𝔽₂)	D₄	1	8	yes	complemented
56	C₇×H(5,𝔽₂)	C₂³	1	12	yes	—
57	C₂×C₇×#41	—	2	16	no	—
58	C₂×C₇×#41	—	2	24	no	—
59	C₂×C₇×#41	—	2	8	no	—
60	C₂×C₇×#41	—	2	24	no	—
61	C₂×C₇×#41	—	2	1	no	—
62	C₂×C₇×#41	—	2	12	no	—
63	C₂×C₇×#41	—	2	12	no	—
64	C₂×C₇×#41	D₄	1	2	yes	—
65	C₂×C₇×#41	C₂³	1	12	yes	—
66	C₂×C₇×#41	C₂³	1	3	yes	—
67	C₂×C₇×#41	C₂³	1	1	yes	characteristic
68	C₂×C₇×#41	C₂³	1	3	yes	complemented
69	C₂²×C₇×D₄	—	2	24	no	—
70	C₂²×C₇×D₄	—	2	12	no	—
71	C₂²×C₇×D₄	—	2	12	no	—
72	C₂²×C₇×D₄	—	2	6	no	—
73	C₂²×C₇×D₄	—	2	3	no	—
74	C₂²×C₇×D₄	D₄	1	6	yes	—
75	C₂²×C₇×D₄	C₂³	1	3	yes	—
76	C₂²×C₇×D₄	C₂³	1	3	yes	complemented
77	C₇×#137	—	2	12	no	—
78	C₇×#137	C₂³	1	8	yes	complemented
79	C₇×AGL(1,ℤ/8)	—	2	24	no	—
80	C₇×AGL(1,ℤ/8)	—	2	12	no	—
81	C₇×AGL(1,ℤ/8)	C₂³	1	24	yes	complemented
82	C₇×#135	—	2	24	no	—
83	C₂×C₇×SD₁₆	—	2	12	no	—
84	C₂×C₇×D₈	—	2	12	no	—
85	C₇×#131	—	2	8	no	—
86	C₂×C₇×M₁₆	C₂³	1	12	yes	complemented
87	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic, complemented
88	C₇×#127	—	2	6	no	—
89	C₇×#127	C₂³	1	3	yes	complemented
90	C₇×#127	C₂³	1	6	yes	complemented
91	C₇×#124	C₂³	1	3	yes	complemented
92	C₇×#124	C₂³	1	3	yes	complemented
93	C₇×#121	—	2	6	no	—
94	C₇×#121	—	2	6	no	—
95	C₇×#121	—	2	6	no	—
96	C₇×#120	—	2	6	no	—
97	C₇×#120	—	2	6	no	—
98	C₄×C₇×Q₈	—	2	1	no	—
99	C₄×C₇×D₄	—	2	3	no	—
100	C₄×C₇×D₄	—	2	6	no	—
101	C₇×#117	C₂³	1	3	yes	complemented
102	C₂×C₄²×C₇	C₂³	1	3	yes	complemented
103	C₇×#104	C₂³	1	8	yes	complemented
104	C₇×#104	C₂³	1	24	yes	complemented
105	C₇×#101	C₂³	1	4	yes	complemented
106	C₇×#100	C₂³	1	12	yes	complemented
107	H(7,𝔽₂)	C₂×C₇	1	1	yes	characteristic, complemented
108	#3585	C₂×C₇	1	2	yes	complemented
109	#3498	C₂×C₇	1	1	yes	characteristic, complemented
110	#3021	C₂×C₇	1	6	yes	complemented
111	#3019	C₂×C₇	1	6	yes	complemented
112	#3018	C₂×C₇	1	6	yes	complemented
113	C₂×#453	C₂×C₇	1	6	yes	complemented
114	#2901	C₂×C₇	1	2	yes	complemented
115	#2888	C₂×C₇	1	1	yes	characteristic, complemented
116	C₂⁴×C₇	—	2	12	no	—
117	C₂⁴×C₇	—	2	3	no	—
118	C₂⁴×C₇	C₂×D₄	1	6	yes	—
119	C₇×#41	—	2	48	no	—
120	C₇×#41	—	2	48	no	—
121	C₇×#41	—	2	24	no	—
122	C₇×#41	—	2	24	no	—
123	C₇×#41	—	2	8	no	—
124	C₇×#41	—	2	8	no	—
125	C₇×#41	—	2	24	no	—
126	C₇×#41	—	2	24	no	—
127	C₇×#41	—	2	8	no	—
128	C₇×#41	—	2	8	no	—
129	C₇×#41	—	2	12	no	—
130	C₇×#41	—	2	12	no	—
131	C₇×#41	—	2	12	no	—
132	C₇×#41	—	2	12	no	—
133	C₇×#41	—	4	4	no	—
134	C₇×#41	C₂×D₄	1	8	yes	—
135	C₇×#41	C₂×D₄	1	24	yes	complemented
136	C₇×#41	C₂⁴	1	8	yes	—
137	C₂×C₇×Q₈	—	2	12	no	—
138	C₂×C₇×Q₈	—	2	8	no	—
139	C₂×C₇×Q₈	—	2	8	no	—
140	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
141	C₂×C₇×Q₈	C₂×D₄	1	12	yes	complemented
142	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
143	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
144	C₂×C₇×D₄	—	2	48	no	—
145	C₂×C₇×D₄	—	2	48	no	—
146	C₂×C₇×D₄	—	2	24	no	—
147	C₂×C₇×D₄	—	2	48	no	—
148	C₂×C₇×D₄	—	2	12	no	—
149	C₂×C₇×D₄	—	2	24	no	—
150	C₂×C₇×D₄	—	2	12	no	—
151	C₂×C₇×D₄	—	2	24	no	—
152	C₂×C₇×D₄	—	2	24	no	—
153	C₂×C₇×D₄	—	2	24	no	—
154	C₂×C₇×D₄	—	2	6	no	—
155	C₂×C₇×D₄	—	2	3	no	—
156	C₂×C₇×D₄	—	2	12	no	—
157	C₂×C₇×D₄	—	2	12	no	—
158	C₂×C₇×D₄	—	2	12	no	—
159	C₂×C₇×D₄	—	2	3	no	—
160	C₂×C₇×D₄	—	4	12	no	—
161	C₂×C₇×D₄	—	8	12	no	—
162	C₂×C₇×D₄	C₂×D₄	1	6	yes	—
163	C₂×C₇×D₄	C₂×D₄	1	6	yes	—
164	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
165	C₂×C₇×D₄	C₂×D₄	1	12	yes	complemented
166	C₂×C₇×D₄	C₂⁴	1	3	yes	—
167	C₂×C₇×D₄	C₂⁴	1	3	yes	—
168	C₂²×C₄×C₇	—	2	12	no	—
169	C₂²×C₄×C₇	—	2	24	no	—
170	C₂²×C₄×C₇	—	2	12	no	—
171	C₂²×C₄×C₇	—	2	3	no	—
172	C₂²×C₄×C₇	—	2	3	no	—
173	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
174	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
175	C₇×Q₁₆	—	2	8	no	—
176	C₇×SD₁₆	—	2	24	no	—
177	C₇×SD₁₆	—	2	8	no	—
178	C₇×D₈	—	2	24	no	—
179	C₇×M₁₆	—	2	12	no	—
180	C₇×M₁₆	C₂⁴	1	8	yes	—
181	C₂×C₇×C₈	—	2	12	no	—
182	C₇×C₄⋊C₄	—	2	3	no	—
183	C₇×C₄⋊C₄	—	2	2	no	—
184	C₇×C₂²⋊C₄	—	2	3	no	—
185	C₇×C₂²⋊C₄	—	2	6	no	—
186	C₇×C₂²⋊C₄	—	2	6	no	—
187	C₄²×C₇	—	2	3	no	—
188	C₄²×C₇	C₂⁴	1	1	yes	characteristic
189	C₄²×C₇	C₂⁴	1	3	yes	—
190	#585	—	2	8	no	—
191	#585	—	2	2	no	—
192	#585	C₂²×C₇	1	2	yes	—
193	#585	C₂²×C₇	1	6	yes	complemented
194	C₂×H(5,𝔽₂)	—	2	8	no	—
195	C₂×H(5,𝔽₂)	—	2	6	no	—
196	C₂×H(5,𝔽₂)	C₂²×C₇	1	1	yes	characteristic
197	C₂×H(5,𝔽₂)	C₂²×C₇	1	6	yes	complemented
198	C₂×H(5,𝔽₂)	C₂²×C₇	1	1	yes	characteristic, complemented
199	#577	—	2	8	no	—
200	#576	—	2	8	no	—
201	#575	C₂²×C₇	1	12	yes	complemented
202	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	12	yes	complemented
203	#568	C₂²×C₇	1	2	yes	complemented
204	#550	—	2	2	no	—
205	D₄²	—	2	6	no	—
206	#535	C₂²×C₇	1	3	yes	complemented
207	#535	C₂²×C₇	1	3	yes	complemented
208	#532	C₂²×C₇	1	3	yes	complemented
209	C₂×#127	C₂²×C₇	1	3	yes	complemented
210	C₄×#41	C₂²×C₇	1	1	yes	characteristic, complemented
211	#456	C₂²×C₇	1	4	yes	complemented
212	#455	C₂²×C₇	1	12	yes	complemented
213	#454	C₂²×C₇	1	12	yes	complemented
214	#453	C₂²×C₇	1	12	yes	complemented
215	#453	C₂²×C₇	1	24	yes	complemented
216	#421	C₂²×C₇	1	12	yes	complemented
217	C₂×#104	C₂²×C₇	1	12	yes	complemented
218	#413	C₂²×C₇	1	6	yes	complemented
219	C₂×#100	C₂²×C₇	1	6	yes	complemented
220	C₂³×C₇	—	2	24	no	—
221	C₂³×C₇	—	2	12	no	—
222	C₂³×C₇	—	2	12	no	—
223	C₂³×C₇	—	2	3	no	—
224	C₂³×C₇	—	4	6	no	—
225	C₂³×C₇	—	8	6	no	—
226	C₂³×C₇	—	8	6	no	—
227	C₂³×C₇	—	16	12	no	—
228	C₂³×C₇	C₂²×D₄	1	6	yes	—
229	C₂³×C₇	C₂²×D₄	1	3	yes	—
230	C₇×Q₈	—	2	8	no	—
231	C₇×Q₈	—	2	12	no	—
232	C₇×Q₈	—	2	2	no	—
233	C₇×Q₈	—	2	2	no	—
234	C₇×Q₈	#120	1	4	yes	—
235	C₇×Q₈	C₂²×D₄	1	8	yes	—
236	C₇×D₄	—	2	48	no	—
237	C₇×D₄	—	2	8	no	—
238	C₇×D₄	—	2	24	no	—
239	C₇×D₄	—	2	12	no	—
240	C₇×D₄	—	2	8	no	—
241	C₇×D₄	—	2	12	no	—
242	C₇×D₄	—	2	12	no	—
243	C₇×D₄	—	2	6	no	—
244	C₇×D₄	—	2	6	no	—
245	C₇×D₄	—	4	12	no	—
246	C₇×D₄	—	4	6	no	—
247	C₇×D₄	—	4	6	no	—
248	C₇×D₄	—	8	12	no	—
249	C₇×D₄	#120	1	12	yes	—
250	C₇×D₄	C₂²×D₄	1	8	yes	—
251	C₂×C₄×C₇	—	2	24	no	—
252	C₂×C₄×C₇	—	2	16	no	—
253	C₂×C₄×C₇	—	2	12	no	—
254	C₂×C₄×C₇	—	2	1	no	—
255	C₂×C₄×C₇	—	2	24	no	—
256	C₂×C₄×C₇	—	2	12	no	—
257	C₂×C₄×C₇	—	2	8	no	—
258	C₂×C₄×C₇	—	2	3	no	—
259	C₂×C₄×C₇	—	2	1	no	—
260	C₂×C₄×C₇	—	4	6	no	—
261	C₂×C₄×C₇	—	4	6	no	—
262	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
264	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
265	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
266	C₇×C₈	—	2	8	no	—
267	2₋¹⁺⁴	—	2	8	no	—
268	2₋¹⁺⁴	—	2	2	no	—
269	2₋¹⁺⁴	—	2	12	no	—
270	2₋¹⁺⁴	C₇×D₄	1	8	yes	complemented
271	2₋¹⁺⁴	C₂³×C₇	1	4	yes	—
272	H(5,𝔽₂)	—	2	8	no	—
273	H(5,𝔽₂)	—	2	48	no	—
274	H(5,𝔽₂)	—	2	8	no	—
275	H(5,𝔽₂)	—	2	24	no	—
276	H(5,𝔽₂)	—	2	6	no	—
277	H(5,𝔽₂)	—	2	12	no	—
278	H(5,𝔽₂)	—	2	12	no	—
279	H(5,𝔽₂)	—	2	12	no	—
280	H(5,𝔽₂)	—	2	8	no	—
281	H(5,𝔽₂)	C₇×D₄	1	8	yes	complemented
282	H(5,𝔽₂)	C₂³×C₇	1	12	yes	—
283	C₂×#41	—	2	16	no	—
284	C₂×#41	—	2	24	no	—
285	C₂×#41	—	2	8	no	—
286	C₂×#41	—	2	24	no	—
287	C₂×#41	—	2	1	no	—
288	C₂×#41	—	2	12	no	—
289	C₂×#41	—	2	12	no	—
290	C₂×#41	C₇×D₄	1	2	yes	—
291	C₂×#41	C₂³×C₇	1	12	yes	—
292	C₂×#41	C₂³×C₇	1	3	yes	—
293	C₂×#41	C₂³×C₇	1	1	yes	characteristic
294	C₂×#41	C₂³×C₇	1	3	yes	complemented
295	C₂²×D₄	—	2	24	no	—
296	C₂²×D₄	—	2	12	no	—
297	C₂²×D₄	—	2	12	no	—
298	C₂²×D₄	—	2	6	no	—
299	C₂²×D₄	—	2	3	no	—
300	C₂²×D₄	C₇×D₄	1	6	yes	—
301	C₂²×D₄	C₂³×C₇	1	3	yes	—
302	C₂²×D₄	C₂³×C₇	1	3	yes	complemented
303	#137	—	2	12	no	—
304	#137	C₂³×C₇	1	8	yes	complemented
305	AGL(1,ℤ/8)	—	2	24	no	—
306	AGL(1,ℤ/8)	—	2	12	no	—
307	AGL(1,ℤ/8)	C₂³×C₇	1	24	yes	complemented
308	#135	—	2	24	no	—
309	C₂×SD₁₆	—	2	12	no	—
310	C₂×D₈	—	2	12	no	—
311	#131	—	2	8	no	—
312	C₂×M₁₆	C₂³×C₇	1	12	yes	complemented
313	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic, complemented
314	#127	—	2	6	no	—
315	#127	C₂³×C₇	1	3	yes	complemented
316	#127	C₂³×C₇	1	6	yes	complemented
317	#124	C₂³×C₇	1	3	yes	complemented
318	#124	C₂³×C₇	1	3	yes	complemented
319	#121	—	2	6	no	—
320	#121	—	2	6	no	—
321	#121	—	2	6	no	—
322	#120	—	2	6	no	—
323	#120	—	2	6	no	—
324	C₄×Q₈	—	2	1	no	—
325	C₄×D₄	—	2	3	no	—
326	C₄×D₄	—	2	6	no	—
327	#117	C₂³×C₇	1	3	yes	complemented
328	C₂×C₄²	C₂³×C₇	1	3	yes	complemented
329	#104	C₂³×C₇	1	8	yes	complemented
330	#104	C₂³×C₇	1	24	yes	complemented
331	#101	C₂³×C₇	1	4	yes	complemented
332	#100	C₂³×C₇	1	12	yes	complemented
333	C₂²×C₇	—	2	6	no	—
334	C₂²×C₇	—	2	8	no	—
335	C₂²×C₇	—	4	3	no	—
336	C₂²×C₇	—	4	6	no	—
337	C₂²×C₇	—	4	6	no	—
338	C₂²×C₇	—	4	3	no	—
339	C₂²×C₇	—	8	12	no	—
340	C₂²×C₇	—	8	12	no	—
341	C₂²×C₇	—	8	24	no	—
342	C₂²×C₇	—	16	6	no	—
343	C₂²×C₇	C₂×#120	1	6	yes	—
344	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
345	C₄×C₇	—	2	2	no	—
346	C₄×C₇	—	2	8	no	—
347	C₄×C₇	—	2	2	no	—
348	C₄×C₇	—	4	3	no	—
349	C₄×C₇	—	4	1	no	—
350	C₄×C₇	C₂×#120	1	6	yes	—
351	C₄×C₇	C₂³×D₄	1	2	yes	—
352	C₂⁴	—	2	12	no	—
353	C₂⁴	—	2	3	no	—
354	C₂⁴	C₂×C₇×D₄	1	6	yes	—
355	#41	—	2	48	no	—
356	#41	—	2	48	no	—
357	#41	—	2	24	no	—
358	#41	—	2	24	no	—
359	#41	—	2	8	no	—
360	#41	—	2	8	no	—
361	#41	—	2	24	no	—
362	#41	—	2	24	no	—
363	#41	—	2	8	no	—
364	#41	—	2	8	no	—
365	#41	—	2	12	no	—
366	#41	—	2	12	no	—
367	#41	—	2	12	no	—
368	#41	—	2	12	no	—
369	#41	—	4	4	no	—
370	#41	C₂×C₇×D₄	1	8	yes	—
371	#41	C₂×C₇×D₄	1	24	yes	complemented
372	#41	C₂⁴×C₇	1	8	yes	—
373	C₂×Q₈	—	2	12	no	—
374	C₂×Q₈	—	2	8	no	—
375	C₂×Q₈	—	2	8	no	—
376	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
377	C₂×Q₈	C₂×C₇×D₄	1	12	yes	complemented
378	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
379	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
380	C₂×D₄	—	2	48	no	—
381	C₂×D₄	—	2	48	no	—
382	C₂×D₄	—	2	24	no	—
383	C₂×D₄	—	2	48	no	—
384	C₂×D₄	—	2	12	no	—
385	C₂×D₄	—	2	24	no	—
386	C₂×D₄	—	2	12	no	—
387	C₂×D₄	—	2	24	no	—
388	C₂×D₄	—	2	24	no	—
389	C₂×D₄	—	2	24	no	—
390	C₂×D₄	—	2	6	no	—
391	C₂×D₄	—	2	3	no	—
392	C₂×D₄	—	2	12	no	—
393	C₂×D₄	—	2	12	no	—
394	C₂×D₄	—	2	12	no	—
395	C₂×D₄	—	2	3	no	—
396	C₂×D₄	—	4	12	no	—
397	C₂×D₄	—	8	12	no	—
398	C₂×D₄	C₂×C₇×D₄	1	6	yes	—
399	C₂×D₄	C₂×C₇×D₄	1	6	yes	—
400	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
401	C₂×D₄	C₂×C₇×D₄	1	12	yes	complemented
402	C₂×D₄	C₂⁴×C₇	1	3	yes	—
403	C₂×D₄	C₂⁴×C₇	1	3	yes	—
404	C₂²×C₄	—	2	12	no	—
405	C₂²×C₄	—	2	24	no	—
406	C₂²×C₄	—	2	12	no	—
407	C₂²×C₄	—	2	3	no	—
408	C₂²×C₄	—	2	3	no	—
409	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
410	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
411	Q₁₆	—	2	8	no	—
412	SD₁₆	—	2	24	no	—
413	SD₁₆	—	2	8	no	—
414	D₈	—	2	24	no	—
415	M₁₆	—	2	12	no	—
416	M₁₆	C₂⁴×C₇	1	8	yes	—
417	C₂×C₈	—	2	12	no	—
418	C₄⋊C₄	—	2	3	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₂²⋊C₄	—	2	3	no	—
421	C₂²⋊C₄	—	2	6	no	—
422	C₂²⋊C₄	—	2	6	no	—
423	C₄²	—	2	3	no	—
424	C₄²	C₂⁴×C₇	1	1	yes	characteristic
425	C₄²	C₂⁴×C₇	1	3	yes	—
426	C₂×C₇	—	2	1	no	—
427	C₂×C₇	—	2	6	no	—
428	C₂×C₇	—	4	6	no	—
429	C₂×C₇	—	4	8	no	—
430	C₂×C₇	—	8	3	no	—
431	C₂×C₇	C₂²×#120	1	1	yes	characteristic
432	C₂³	—	2	24	no	—
433	C₂³	—	2	12	no	—
434	C₂³	—	2	12	no	—
435	C₂³	—	2	3	no	—
436	C₂³	—	4	6	no	—
437	C₂³	—	8	6	no	—
438	C₂³	—	8	6	no	—
439	C₂³	—	16	12	no	—
440	C₂³	C₂²×C₇×D₄	1	6	yes	—
441	C₂³	C₂²×C₇×D₄	1	3	yes	—
442	Q₈	—	2	8	no	—
443	Q₈	—	2	2	no	—
444	Q₈	—	2	12	no	—
445	Q₈	—	2	2	no	—
446	Q₈	C₇×#120	1	4	yes	—
447	Q₈	C₂²×C₇×D₄	1	8	yes	—
448	D₄	—	2	48	no	—
449	D₄	—	2	8	no	—
450	D₄	—	2	8	no	—
451	D₄	—	2	24	no	—
452	D₄	—	2	12	no	—
453	D₄	—	2	6	no	—
454	D₄	—	2	6	no	—
455	D₄	—	2	12	no	—
456	D₄	—	2	12	no	—
457	D₄	—	4	12	no	—
458	D₄	—	4	6	no	—
459	D₄	—	4	6	no	—
460	D₄	—	8	12	no	—
461	D₄	C₇×#120	1	12	yes	—
462	D₄	C₂²×C₇×D₄	1	8	yes	—
463	C₂×C₄	—	2	24	no	—
464	C₂×C₄	—	2	16	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	12	no	—
467	C₂×C₄	—	2	8	no	—
468	C₂×C₄	—	2	24	no	—
469	C₂×C₄	—	2	12	no	—
470	C₂×C₄	—	2	3	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	4	6	no	—
473	C₂×C₄	—	4	6	no	—
474	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
475	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
476	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
477	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
478	C₈	—	2	8	no	—
479	C₇	#60388	1	1	yes	characteristic, direct product factor, Sylow
480	C₂²	—	2	6	no	—
481	C₂²	—	2	8	no	—
482	C₂²	—	4	3	no	—
483	C₂²	—	4	6	no	—
484	C₂²	—	4	6	no	—
485	C₂²	—	4	3	no	—
486	C₂²	—	8	12	no	—
487	C₂²	—	8	12	no	—
488	C₂²	—	8	24	no	—
489	C₂²	—	16	6	no	—
490	C₂²	C₂×C₇×#120	1	6	yes	—
491	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
492	C₄	—	2	2	no	—
493	C₄	—	2	8	no	—
494	C₄	—	2	2	no	—
495	C₄	—	4	3	no	—
496	C₄	—	4	1	no	—
497	C₄	C₂×C₇×#120	1	6	yes	—
498	C₄	C₂³×C₇×D₄	1	2	yes	—
499	C₂	—	2	1	no	—
500	C₂	—	2	6	no	—
501	C₂	—	4	6	no	—
502	C₂	—	4	8	no	—
503	C₂	—	8	3	no	—
504	C₂	C₂²×C₇×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	C₂²×#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×H(7,𝔽₂)	C₂	semidirect
2. C₇×#3585	C₂	semidirect
3. C₇×#3498	C₂	semidirect
4. C₇×#3021	C₂	semidirect
5. C₇×#3019	C₂	semidirect
6. C₇×#3018	C₂	semidirect
7. C₂×C₇×#453	C₂	semidirect
8. C₇×#2901	C₂	semidirect
9. C₇×#2888	C₂	semidirect
13. C₇×#585	C₂²	semidirect
17. C₂×C₇×H(5,𝔽₂)	C₂²	semidirect
18. C₂×C₇×H(5,𝔽₂)	C₂²	semidirect
21. C₇×#575	C₂²	semidirect
22. C₂×C₇×AGL(1,ℤ/8)	C₂²	semidirect
23. C₇×#568	C₂²	semidirect
26. C₇×#535	C₂²	semidirect
27. C₇×#535	C₂²	semidirect
28. C₇×#532	C₂²	semidirect
29. C₂×C₇×#127	C₂²	semidirect
30. C₄×C₇×#41	C₂²	semidirect
31. C₇×#456	C₂²	semidirect
32. C₇×#455	C₂²	semidirect
33. C₇×#454	C₂²	semidirect
34. C₇×#453	C₂²	semidirect
35. C₇×#453	C₂²	semidirect
36. C₇×#421	C₂²	semidirect
37. C₂×C₇×#104	C₂²	semidirect
38. C₇×#413	C₂²	semidirect
39. C₂×C₇×#100	C₂²	semidirect
40. #60388	C₇	direct (N × H)
44. C₇×2₋¹⁺⁴	D₄	semidirect
55. C₇×H(5,𝔽₂)	D₄	semidirect
68. C₂×C₇×#41	C₂³	semidirect
76. C₂²×C₇×D₄	C₂³	semidirect
78. C₇×#137	C₂³	semidirect
81. C₇×AGL(1,ℤ/8)	C₂³	semidirect
86. C₂×C₇×M₁₆	C₂³	semidirect
87. C₇×C₄⋊Q₈	C₂³	semidirect
89. C₇×#127	C₂³	semidirect
90. C₇×#127	C₂³	semidirect
91. C₇×#124	C₂³	semidirect
92. C₇×#124	C₂³	semidirect
101. C₇×#117	C₂³	semidirect
102. C₂×C₄²×C₇	C₂³	semidirect
103. C₇×#104	C₂³	semidirect
104. C₇×#104	C₂³	semidirect
105. C₇×#101	C₂³	semidirect
106. C₇×#100	C₂³	semidirect
107. H(7,𝔽₂)	C₂×C₇	semidirect
108. #3585	C₂×C₇	semidirect
109. #3498	C₂×C₇	semidirect
110. #3021	C₂×C₇	semidirect
111. #3019	C₂×C₇	semidirect
112. #3018	C₂×C₇	semidirect
113. C₂×#453	C₂×C₇	semidirect
114. #2901	C₂×C₇	semidirect
115. #2888	C₂×C₇	semidirect
135. C₇×#41	C₂×D₄	semidirect
141. C₂×C₇×Q₈	C₂×D₄	semidirect
165. C₂×C₇×D₄	C₂×D₄	semidirect
193. #585	C₂²×C₇	semidirect
197. C₂×H(5,𝔽₂)	C₂²×C₇	semidirect
198. C₂×H(5,𝔽₂)	C₂²×C₇	semidirect
201. #575	C₂²×C₇	semidirect
202. C₂×AGL(1,ℤ/8)	C₂²×C₇	semidirect
203. #568	C₂²×C₇	semidirect
206. #535	C₂²×C₇	semidirect
207. #535	C₂²×C₇	semidirect
208. #532	C₂²×C₇	semidirect
209. C₂×#127	C₂²×C₇	semidirect
210. C₄×#41	C₂²×C₇	semidirect
211. #456	C₂²×C₇	semidirect
212. #455	C₂²×C₇	semidirect
213. #454	C₂²×C₇	semidirect
214. #453	C₂²×C₇	semidirect
215. #453	C₂²×C₇	semidirect
216. #421	C₂²×C₇	semidirect
217. C₂×#104	C₂²×C₇	semidirect
218. #413	C₂²×C₇	semidirect
219. C₂×#100	C₂²×C₇	semidirect
270. 2₋¹⁺⁴	C₇×D₄	semidirect
281. H(5,𝔽₂)	C₇×D₄	semidirect
294. C₂×#41	C₂³×C₇	semidirect
302. C₂²×D₄	C₂³×C₇	semidirect
304. #137	C₂³×C₇	semidirect
307. AGL(1,ℤ/8)	C₂³×C₇	semidirect
312. C₂×M₁₆	C₂³×C₇	semidirect
313. C₄⋊Q₈	C₂³×C₇	semidirect
315. #127	C₂³×C₇	semidirect
316. #127	C₂³×C₇	semidirect
317. #124	C₂³×C₇	semidirect
318. #124	C₂³×C₇	semidirect
327. #117	C₂³×C₇	semidirect
328. C₂×C₄²	C₂³×C₇	semidirect
329. #104	C₂³×C₇	semidirect
330. #104	C₂³×C₇	semidirect
331. #101	C₂³×C₇	semidirect
332. #100	C₂³×C₇	semidirect
371. #41	C₂×C₇×D₄	semidirect
377. C₂×Q₈	C₂×C₇×D₄	semidirect
401. C₂×D₄	C₂×C₇×D₄	semidirect
479. C₇	#60388	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×H(7,𝔽₂)	2
C₇×#3585	2
C₇×#3498	2
C₇×#3021	2
C₇×#3019	2
C₇×#3018	2
C₂×C₇×#453	2
C₇×#2901	2
C₇×#2888	2
#60388	7

Subgroups — C7×#60388 #13237604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60388 #13237604