Subgroups — C₂×C₇×#3304 #13238338

Order 1792.

2198 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#550	C₂	1	1	yes	maximal, characteristic
3	C₇×#3304	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₇×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×#482	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#475	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#465	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×#460	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×#452	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#416	C₂	1	2	yes	maximal, complemented
13	C₂²×C₇×#41	C₂²	1	2	yes	—
14	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₇×SD₁₆	C₂²	1	2	yes	—
16	C₂³×C₇×C₈	C₂²	1	2	yes	—
17	C₇×#556	—	2	4	no	—
18	C₇×#556	C₂²	1	8	yes	complemented
19	C₇×#550	—	2	4	no	—
20	C₇×#550	C₂²	1	8	yes	—
21	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₇×#127	—	2	1	no	—
23	C₂×C₇×#127	C₂²	1	1	yes	characteristic
24	C₂×C₇×#126	—	2	2	no	—
25	C₂×C₇×#124	—	2	1	no	—
26	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂×C₇×#121	—	2	2	no	—
28	C₂×C₇×#121	C₂²	1	2	yes	—
29	C₂×C₄×C₇×Q₈	—	2	1	no	—
30	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₇×D₄	—	2	1	no	—
33	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
34	C₂×C₇×#117	C₂²	1	2	yes	—
35	C₇×#486	C₂²	1	8	yes	complemented
36	C₇×#482	C₂²	1	16	yes	complemented
37	C₇×#475	C₂²	1	8	yes	complemented
38	C₇×#465	C₂²	1	16	yes	complemented
39	C₇×#460	C₂²	1	8	yes	complemented
40	C₇×#452	C₂²	1	16	yes	complemented
41	C₄×C₇×Q₁₆	C₂²	1	8	yes	complemented
42	C₇×C₈×D₄	C₂²	1	8	yes	complemented
43	C₂×C₇×#107	C₂²	1	1	yes	characteristic
44	C₂×C₇×#106	C₂²	1	2	yes	—
45	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₇×#416	C₂²	1	16	yes	complemented
47	C₂×C₇×#103	C₂²	1	1	yes	characteristic
48	C₂×C₇×#103	C₂²	1	1	yes	characteristic
49	C₂×C₇×#103	C₂²	1	2	yes	—
50	C₂×C₇×#102	C₂²	1	2	yes	—
51	C₂×C₇×#102	C₂²	1	2	yes	—
52	C₂×C₇×#102	C₂²	1	2	yes	—
53	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
54	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
55	C₂×#3304	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂×C₇×#41	—	2	4	no	—
57	C₂×C₇×#41	—	2	8	no	—
58	C₂×C₇×#41	—	4	4	no	—
59	C₂×C₇×#41	C₂³	1	8	yes	—
60	C₂²×C₇×Q₈	—	2	1	no	—
61	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
62	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
63	C₂²×C₇×D₄	—	2	2	no	—
64	C₂²×C₇×D₄	—	2	2	no	—
65	C₂²×C₇×D₄	—	2	1	no	—
66	C₂²×C₇×D₄	—	2	1	no	—
67	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
68	C₂²×C₇×D₄	C₂³	1	2	yes	—
69	C₂³×C₄×C₇	—	2	2	no	—
70	C₂³×C₄×C₇	C₂³	1	2	yes	—
71	C₂×C₇×Q₁₆	—	2	4	no	—
72	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
73	C₂×C₇×SD₁₆	—	4	4	no	—
74	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
75	C₂²×C₇×C₈	—	4	4	no	—
76	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
77	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
78	C₂²×C₇×C₈	C₂³	1	2	yes	—
79	C₂²×C₇×C₈	C₂³	1	8	yes	—
80	C₇×C₄⋊Q₈	—	2	2	no	—
81	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
82	C₇×#127	—	2	8	no	—
83	C₇×#127	—	2	2	no	—
84	C₇×#127	C₂³	1	4	yes	—
85	C₇×#126	—	2	16	no	—
86	C₇×#124	—	2	8	no	—
87	C₇×C₂²⋊Q₈	—	2	4	no	—
88	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
89	C₇×#121	—	2	4	no	—
90	C₇×#121	—	2	16	no	—
91	C₇×#121	C₂³	1	8	yes	—
92	C₄×C₇×Q₈	—	2	8	no	—
93	C₄×C₇×Q₈	—	2	2	no	—
94	C₄×C₇×Q₈	—	2	2	no	—
95	C₄×C₇×Q₈	C₂³	1	4	yes	—
96	C₄×C₇×Q₈	C₂³	1	4	yes	—
97	C₄×C₇×D₄	—	2	8	no	—
98	C₄×C₇×D₄	D₄	1	4	yes	—
99	C₄×C₇×D₄	C₂³	1	4	yes	—
100	C₇×#117	—	2	4	no	—
101	C₇×#117	C₂³	1	8	yes	—
102	C₂×C₇×C₄⋊C₄	—	2	2	no	—
103	C₂×C₇×C₄⋊C₄	—	2	1	no	—
104	C₂×C₇×C₄⋊C₄	—	2	1	no	—
105	C₂×C₇×C₄⋊C₄	—	2	1	no	—
106	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
107	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
108	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
109	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
110	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
111	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
112	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
113	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
114	C₂×C₄²×C₇	—	2	2	no	—
115	C₂×C₄²×C₇	—	2	1	no	—
116	C₂×C₄²×C₇	—	2	1	no	—
117	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
118	C₇×#107	C₂³	1	4	yes	—
119	C₇×#106	C₂³	1	8	yes	—
120	C₇×C₄⋊C₈	C₂³	1	4	yes	—
121	C₇×#103	C₂³	1	8	yes	—
122	C₇×#103	C₂³	1	4	yes	—
123	C₇×#103	C₂³	1	4	yes	—
124	C₇×#102	C₂³	1	8	yes	—
125	C₇×#102	C₂³	1	8	yes	—
126	C₇×#102	C₂³	1	8	yes	—
127	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
128	C₄×C₇×C₈	C₂³	1	4	yes	—
129	C₂×#556	C₂×C₇	1	1	yes	characteristic, complemented
130	C₂×#550	C₂×C₇	1	1	yes	characteristic
131	#3304	C₂×C₇	1	16	yes	direct product factor
132	C₂×#486	C₂×C₇	1	1	yes	characteristic, complemented
133	C₂×#482	C₂×C₇	1	2	yes	complemented
134	C₂×#475	C₂×C₇	1	1	yes	characteristic, complemented
135	C₂×#465	C₂×C₇	1	2	yes	complemented
136	C₂×#460	C₂×C₇	1	1	yes	characteristic, complemented
137	C₂×#452	C₂×C₇	1	2	yes	complemented
138	C₂×C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
139	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic, complemented
140	C₂×#416	C₂×C₇	1	2	yes	complemented
141	C₂⁴×C₇	—	2	2	no	—
142	C₂⁴×C₇	C₂×D₄	1	2	yes	—
143	C₇×#41	—	2	16	no	—
144	C₇×#41	—	4	8	no	—
145	C₇×#41	—	4	16	no	—
146	C₂×C₇×Q₈	—	2	1	no	—
147	C₂×C₇×Q₈	—	2	1	no	—
148	C₂×C₇×Q₈	—	2	4	no	—
149	C₂×C₇×Q₈	—	2	2	no	—
150	C₂×C₇×Q₈	—	2	2	no	—
151	C₂×C₇×Q₈	—	2	2	no	—
152	C₂×C₇×Q₈	—	4	4	no	—
153	C₂×C₇×Q₈	#41	1	4	yes	—
154	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
155	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
156	C₂×C₇×D₄	—	2	8	no	—
157	C₂×C₇×D₄	—	2	8	no	—
158	C₂×C₇×D₄	—	2	4	no	—
159	C₂×C₇×D₄	—	2	4	no	—
160	C₂×C₇×D₄	—	2	2	no	—
161	C₂×C₇×D₄	—	2	4	no	—
162	C₂×C₇×D₄	—	2	4	no	—
163	C₂×C₇×D₄	—	2	4	no	—
164	C₂×C₇×D₄	—	4	8	no	—
165	C₂×C₇×D₄	—	4	4	no	—
166	C₂×C₇×D₄	—	8	2	no	—
167	C₂×C₇×D₄	—	8	4	no	—
168	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
169	C₂×C₇×D₄	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₇	—	2	1	no	—
171	C₂²×C₄×C₇	—	2	4	no	—
172	C₂²×C₄×C₇	—	2	4	no	—
173	C₂²×C₄×C₇	—	2	1	no	—
174	C₂²×C₄×C₇	—	2	1	no	—
175	C₂²×C₄×C₇	—	2	2	no	—
176	C₂²×C₄×C₇	—	2	1	no	—
177	C₂²×C₄×C₇	—	2	8	no	—
178	C₂²×C₄×C₇	—	4	8	no	—
179	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
180	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
184	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
185	C₇×Q₁₆	—	2	8	no	—
186	C₇×SD₁₆	—	4	8	no	—
187	C₂×C₇×C₈	—	2	2	no	—
188	C₂×C₇×C₈	—	2	2	no	—
189	C₂×C₇×C₈	—	2	4	no	—
190	C₂×C₇×C₈	—	4	8	no	—
191	C₂×C₇×C₈	C₂⁴	1	2	yes	—
192	C₂×C₇×C₈	C₂⁴	1	2	yes	—
193	C₂×C₇×C₈	C₂⁴	1	4	yes	—
194	C₇×C₄⋊C₄	—	2	8	no	—
195	C₇×C₄⋊C₄	—	2	4	no	—
196	C₇×C₄⋊C₄	—	2	4	no	—
197	C₇×C₄⋊C₄	—	2	4	no	—
198	C₇×C₄⋊C₄	—	2	1	no	—
199	C₇×C₄⋊C₄	—	2	2	no	—
200	C₇×C₄⋊C₄	—	2	1	no	—
201	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
202	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
203	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
204	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
205	C₇×C₂²⋊C₄	—	2	8	no	—
206	C₇×C₂²⋊C₄	—	2	8	no	—
207	C₇×C₂²⋊C₄	—	2	8	no	—
208	C₇×C₂²⋊C₄	C₂×D₄	1	8	yes	—
209	C₄²×C₇	—	2	8	no	—
210	C₄²×C₇	—	2	4	no	—
211	C₄²×C₇	—	2	4	no	—
212	C₄²×C₇	C₂×D₄	1	2	yes	—
213	C₄²×C₇	C₂⁴	1	2	yes	—
214	C₂²×#41	C₂²×C₇	1	2	yes	—
215	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
216	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
217	C₂³×C₈	C₂²×C₇	1	2	yes	—
218	#556	—	2	4	no	—
219	#556	C₂²×C₇	1	8	yes	complemented
220	#550	—	2	4	no	—
221	#550	C₂²×C₇	1	8	yes	—
222	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
223	C₂×#127	—	2	1	no	—
224	C₂×#127	C₂²×C₇	1	1	yes	characteristic
225	C₂×#126	—	2	2	no	—
226	C₂×#124	—	2	1	no	—
227	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
228	C₂×#121	—	2	2	no	—
229	C₂×#121	C₂²×C₇	1	2	yes	—
230	C₂×C₄×Q₈	—	2	1	no	—
231	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
232	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
233	C₂×C₄×D₄	—	2	1	no	—
234	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
235	C₂×#117	C₂²×C₇	1	2	yes	—
236	#486	C₂²×C₇	1	8	yes	complemented
237	#482	C₂²×C₇	1	16	yes	complemented
238	#475	C₂²×C₇	1	8	yes	complemented
239	#465	C₂²×C₇	1	16	yes	complemented
240	#460	C₂²×C₇	1	8	yes	complemented
241	#452	C₂²×C₇	1	16	yes	complemented
242	C₄×Q₁₆	C₂²×C₇	1	8	yes	complemented
243	C₈×D₄	C₂²×C₇	1	8	yes	complemented
244	C₂×#107	C₂²×C₇	1	1	yes	characteristic
245	C₂×#106	C₂²×C₇	1	2	yes	—
246	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
247	#416	C₂²×C₇	1	16	yes	complemented
248	C₂×#103	C₂²×C₇	1	2	yes	—
249	C₂×#103	C₂²×C₇	1	1	yes	characteristic
250	C₂×#103	C₂²×C₇	1	1	yes	characteristic
251	C₂×#102	C₂²×C₇	1	2	yes	—
252	C₂×#102	C₂²×C₇	1	2	yes	—
253	C₂×#102	C₂²×C₇	1	2	yes	—
254	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
255	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
256	C₂³×C₇	—	2	4	no	—
257	C₂³×C₇	—	2	4	no	—
258	C₂³×C₇	—	2	4	no	—
259	C₂³×C₇	—	2	4	no	—
260	C₂³×C₇	—	4	4	no	—
261	C₂³×C₇	—	4	4	no	—
262	C₂³×C₇	—	4	4	no	—
263	C₂³×C₇	C₂²×D₄	1	4	yes	—
264	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
265	C₇×Q₈	—	2	4	no	—
266	C₇×Q₈	—	2	2	no	—
267	C₇×Q₈	—	2	2	no	—
268	C₇×Q₈	—	2	2	no	—
269	C₇×Q₈	—	4	2	no	—
270	C₇×Q₈	—	4	8	no	—
271	C₇×Q₈	C₂×#41	1	4	yes	—
272	C₇×D₄	—	2	8	no	—
273	C₇×D₄	—	2	4	no	—
274	C₇×D₄	—	2	4	no	—
275	C₇×D₄	—	4	16	no	—
276	C₇×D₄	—	4	4	no	—
277	C₇×D₄	—	4	8	no	—
278	C₇×D₄	—	8	4	no	—
279	C₇×D₄	—	8	8	no	—
280	C₂×C₄×C₇	—	2	4	no	—
281	C₂×C₄×C₇	—	2	4	no	—
282	C₂×C₄×C₇	—	2	2	no	—
283	C₂×C₄×C₇	—	2	2	no	—
284	C₂×C₄×C₇	—	2	8	no	—
285	C₂×C₄×C₇	—	2	4	no	—
286	C₂×C₄×C₇	—	2	4	no	—
287	C₂×C₄×C₇	—	2	2	no	—
288	C₂×C₄×C₇	—	2	2	no	—
289	C₂×C₄×C₇	—	2	4	no	—
290	C₂×C₄×C₇	—	2	2	no	—
291	C₂×C₄×C₇	—	4	16	no	—
292	C₂×C₄×C₇	—	4	2	no	—
293	C₂×C₄×C₇	—	4	4	no	—
294	C₂×C₄×C₇	—	4	2	no	—
295	C₂×C₄×C₇	—	4	2	no	—
296	C₂×C₄×C₇	#135	1	4	yes	—
297	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
298	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
299	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
301	C₂×C₄×C₇	C₂×#41	1	2	yes	—
302	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
303	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
304	C₇×C₈	—	2	2	no	—
305	C₇×C₈	—	2	2	no	—
306	C₇×C₈	—	2	4	no	—
307	C₂×#41	—	2	4	no	—
308	C₂×#41	—	2	8	no	—
309	C₂×#41	—	4	4	no	—
310	C₂×#41	C₂³×C₇	1	8	yes	—
311	C₂²×Q₈	—	2	1	no	—
312	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
313	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
314	C₂²×D₄	—	2	2	no	—
315	C₂²×D₄	—	2	2	no	—
316	C₂²×D₄	—	2	1	no	—
317	C₂²×D₄	—	2	1	no	—
318	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
319	C₂²×D₄	C₂³×C₇	1	2	yes	—
320	C₂³×C₄	—	2	2	no	—
321	C₂³×C₄	C₂³×C₇	1	2	yes	—
322	C₂×Q₁₆	—	2	4	no	—
323	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
324	C₂×SD₁₆	—	4	4	no	—
325	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
326	C₂²×C₈	—	4	4	no	—
327	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
328	C₂²×C₈	C₂³×C₇	1	2	yes	—
329	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₇	1	8	yes	—
331	C₄⋊Q₈	—	2	2	no	—
332	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
333	#127	—	2	8	no	—
334	#127	—	2	2	no	—
335	#127	C₂³×C₇	1	4	yes	—
336	#126	—	2	16	no	—
337	#124	—	2	8	no	—
338	C₂²⋊Q₈	—	2	4	no	—
339	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
340	#121	—	2	4	no	—
341	#121	—	2	16	no	—
342	#121	C₂³×C₇	1	8	yes	—
343	C₄×Q₈	—	2	8	no	—
344	C₄×Q₈	—	2	2	no	—
345	C₄×Q₈	—	2	2	no	—
346	C₄×Q₈	C₂³×C₇	1	4	yes	—
347	C₄×Q₈	C₂³×C₇	1	4	yes	—
348	C₄×D₄	—	2	8	no	—
349	C₄×D₄	C₇×D₄	1	4	yes	—
350	C₄×D₄	C₂³×C₇	1	4	yes	—
351	#117	—	2	4	no	—
352	#117	C₂³×C₇	1	8	yes	—
353	C₂×C₄⋊C₄	—	2	2	no	—
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄⋊C₄	—	2	1	no	—
356	C₂×C₄⋊C₄	—	2	1	no	—
357	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
358	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
359	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
360	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₂²⋊C₄	—	2	2	no	—
364	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
365	C₂×C₄²	—	2	2	no	—
366	C₂×C₄²	—	2	1	no	—
367	C₂×C₄²	—	2	1	no	—
368	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
369	#107	C₂³×C₇	1	4	yes	—
370	#106	C₂³×C₇	1	8	yes	—
371	C₄⋊C₈	C₂³×C₇	1	4	yes	—
372	#103	C₂³×C₇	1	4	yes	—
373	#103	C₂³×C₇	1	8	yes	—
374	#103	C₂³×C₇	1	4	yes	—
375	#102	C₂³×C₇	1	8	yes	—
376	#102	C₂³×C₇	1	8	yes	—
377	#102	C₂³×C₇	1	8	yes	—
378	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
379	C₄×C₈	C₂³×C₇	1	4	yes	—
380	C₂²×C₇	—	2	4	no	—
381	C₂²×C₇	—	2	4	no	—
382	C₂²×C₇	—	2	4	no	—
383	C₂²×C₇	—	4	8	no	—
384	C₂²×C₇	—	4	4	no	—
385	C₂²×C₇	—	4	4	no	—
386	C₂²×C₇	—	4	4	no	—
387	C₂²×C₇	—	8	8	no	—
388	C₂²×C₇	#546	1	2	yes	—
389	C₂²×C₇	C₂×#135	1	2	yes	—
390	C₂²×C₇	#577	1	2	yes	—
391	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
392	C₄×C₇	—	2	4	no	—
393	C₄×C₇	—	2	4	no	—
394	C₄×C₇	—	2	2	no	—
395	C₄×C₇	—	4	2	no	—
396	C₄×C₇	—	4	4	no	—
397	C₄×C₇	—	4	2	no	—
398	C₄×C₇	—	4	2	no	—
399	C₄×C₇	#546	1	2	yes	—
400	C₄×C₇	C₂×#135	1	4	yes	—
401	C₄×C₇	C₂²×#41	1	1	yes	characteristic
402	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
403	C₂⁴	—	2	2	no	—
404	C₂⁴	C₂×C₇×D₄	1	2	yes	—
405	#41	—	2	16	no	—
406	#41	—	4	8	no	—
407	#41	—	4	16	no	—
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	2	1	no	—
410	C₂×Q₈	—	2	4	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	—	2	2	no	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	—	4	4	no	—
415	C₂×Q₈	C₇×#41	1	4	yes	—
416	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
417	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	8	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	4	8	no	—
427	C₂×D₄	—	4	4	no	—
428	C₂×D₄	—	8	2	no	—
429	C₂×D₄	—	8	4	no	—
430	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
431	C₂×D₄	C₂⁴×C₇	1	4	yes	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	8	no	—
440	C₂²×C₄	—	4	8	no	—
441	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
442	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
443	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
444	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
446	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
447	Q₁₆	—	2	8	no	—
448	SD₁₆	—	4	8	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	2	4	no	—
452	C₂×C₈	—	4	8	no	—
453	C₂×C₈	C₂⁴×C₇	1	2	yes	—
454	C₂×C₈	C₂⁴×C₇	1	2	yes	—
455	C₂×C₈	C₂⁴×C₇	1	4	yes	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
464	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
465	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
466	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	C₂×C₇×D₄	1	8	yes	—
471	C₄²	—	2	8	no	—
472	C₄²	—	2	4	no	—
473	C₄²	—	2	4	no	—
474	C₄²	C₂×C₇×D₄	1	2	yes	—
475	C₄²	C₂⁴×C₇	1	2	yes	—
476	C₂×C₇	—	4	4	no	—
477	C₂×C₇	—	8	4	no	—
478	C₂×C₇	#3304	1	4	yes	direct product factor
479	C₂×C₇	C₂×#546	1	1	yes	characteristic
480	C₂×C₇	C₂²×#135	1	1	yes	characteristic
481	C₂×C₇	C₂×#577	1	1	yes	characteristic
482	C₂³	—	2	4	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	2	4	no	—
485	C₂³	—	2	4	no	—
486	C₂³	—	4	4	no	—
487	C₂³	—	4	4	no	—
488	C₂³	—	4	4	no	—
489	C₂³	C₂²×C₇×D₄	1	4	yes	—
490	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
491	Q₈	—	2	2	no	—
492	Q₈	—	2	4	no	—
493	Q₈	—	2	2	no	—
494	Q₈	—	2	2	no	—
495	Q₈	—	4	2	no	—
496	Q₈	—	4	8	no	—
497	Q₈	C₂×C₇×#41	1	4	yes	—
498	D₄	—	2	4	no	—
499	D₄	—	2	4	no	—
500	D₄	—	2	8	no	—
501	D₄	—	4	16	no	—
502	D₄	—	4	4	no	—
503	D₄	—	4	8	no	—
504	D₄	—	8	4	no	—
505	D₄	—	8	8	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	4	16	no	—
518	C₂×C₄	—	4	2	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	C₇×#135	1	4	yes	—
523	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
524	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
525	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
526	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
527	C₂×C₄	C₂×C₇×#41	1	2	yes	—
528	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
529	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
530	C₈	—	2	2	no	—
531	C₈	—	2	2	no	—
532	C₈	—	2	4	no	—
533	C₇	C₂×#3304	1	1	yes	characteristic, direct product factor, Sylow
534	C₂²	—	2	4	no	—
535	C₂²	—	2	4	no	—
536	C₂²	—	2	4	no	—
537	C₂²	—	4	8	no	—
538	C₂²	—	4	4	no	—
539	C₂²	—	4	4	no	—
540	C₂²	—	4	4	no	—
541	C₂²	—	8	8	no	—
542	C₂²	C₇×#546	1	2	yes	—
543	C₂²	C₂×C₇×#135	1	2	yes	—
544	C₂²	C₇×#577	1	2	yes	—
545	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
546	C₄	—	2	2	no	—
547	C₄	—	2	4	no	—
548	C₄	—	2	4	no	—
549	C₄	—	4	4	no	—
550	C₄	—	4	2	no	—
551	C₄	—	4	2	no	—
552	C₄	—	4	2	no	—
553	C₄	C₇×#546	1	2	yes	—
554	C₄	C₂×C₇×#135	1	4	yes	—
555	C₄	C₂²×C₇×#41	1	1	yes	characteristic
556	C₄	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
557	C₂	—	4	4	no	—
558	C₂	—	8	4	no	—
559	C₂	C₇×#3304	1	4	yes	direct product factor
560	C₂	C₂×C₇×#546	1	1	yes	characteristic
561	C₂	C₂²×C₇×#135	1	1	yes	characteristic
562	C₂	C₂×C₇×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂×C₇×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#556	C₂	semidirect
3. C₇×#3304	C₂	direct (N × H)
4. C₂×C₇×#486	C₂	semidirect
5. C₂×C₇×#482	C₂	semidirect
6. C₂×C₇×#475	C₂	semidirect
7. C₂×C₇×#465	C₂	semidirect
8. C₂×C₇×#460	C₂	semidirect
9. C₂×C₇×#452	C₂	semidirect
10. C₂×C₄×C₇×Q₁₆	C₂	semidirect
11. C₂×C₇×C₈×D₄	C₂	semidirect
12. C₂×C₇×#416	C₂	semidirect
18. C₇×#556	C₂²	semidirect
35. C₇×#486	C₂²	semidirect
36. C₇×#482	C₂²	semidirect
37. C₇×#475	C₂²	semidirect
38. C₇×#465	C₂²	semidirect
39. C₇×#460	C₂²	semidirect
40. C₇×#452	C₂²	semidirect
41. C₄×C₇×Q₁₆	C₂²	semidirect
42. C₇×C₈×D₄	C₂²	semidirect
46. C₇×#416	C₂²	semidirect
55. C₂×#3304	C₇	direct (N × H)
129. C₂×#556	C₂×C₇	semidirect
131. #3304	C₂×C₇	direct (N × H)
132. C₂×#486	C₂×C₇	semidirect
133. C₂×#482	C₂×C₇	semidirect
134. C₂×#475	C₂×C₇	semidirect
135. C₂×#465	C₂×C₇	semidirect
136. C₂×#460	C₂×C₇	semidirect
137. C₂×#452	C₂×C₇	semidirect
138. C₂×C₄×Q₁₆	C₂×C₇	semidirect
139. C₂×C₈×D₄	C₂×C₇	semidirect
140. C₂×#416	C₂×C₇	semidirect
219. #556	C₂²×C₇	semidirect
236. #486	C₂²×C₇	semidirect
237. #482	C₂²×C₇	semidirect
238. #475	C₂²×C₇	semidirect
239. #465	C₂²×C₇	semidirect
240. #460	C₂²×C₇	semidirect
241. #452	C₂²×C₇	semidirect
242. C₄×Q₁₆	C₂²×C₇	semidirect
243. C₈×D₄	C₂²×C₇	semidirect
247. #416	C₂²×C₇	semidirect
478. C₂×C₇	#3304	direct (N × H)
533. C₇	C₂×#3304	direct (N × H)
559. C₂	C₇×#3304	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#556	2
C₂×C₇×#550	2
C₇×#3304	2
C₂×C₇×#486	2
C₂×C₇×#482	2
C₂×C₇×#475	2
C₂×C₇×#465	2
C₂×C₇×#460	2
C₂×C₇×#452	2
C₂×C₄×C₇×Q₁₆	2
C₂×C₇×C₈×D₄	2
C₂×C₇×#416	2
C₂×#3304	7

Subgroups — C2×C7×#3304 #13238338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3304 #13238338