Subgroups — C₂²×C₇×#542 #13239894

Order 1792.

7974 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#542	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×#126	C₂	1	2	yes	maximal, complemented
4	C₂²×C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₇×#124	C₂	1	2	yes	maximal, complemented
6	C₂²×C₇×#123	C₂	1	2	yes	maximal, complemented
7	C₂²×C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₇×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
9	C₂²×C₇×#121	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic
14	C₂³×C₇×Q₈	C₂²	1	2	yes	—
15	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
16	C₂³×C₇×D₄	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₇	C₂²	1	2	yes	—
19	C₇×#542	C₂²	1	256	yes	direct product factor
20	C₂×C₇×C₄⋊Q₈	C₂²	1	24	yes	complemented
21	C₂×C₇×#126	C₂²	1	48	yes	complemented
22	C₂×C₇×#124	C₂²	1	24	yes	complemented
23	C₂×C₇×#124	C₂²	1	48	yes	complemented
24	C₂×C₇×#123	C₂²	1	48	yes	complemented
25	C₂×C₇×C₂²⋊Q₈	C₂²	1	48	yes	complemented
26	C₂×C₇×C₂²⋊Q₈	C₂²	1	24	yes	complemented
27	C₂×C₇×#121	C₂²	1	24	yes	—
28	C₂×C₄×C₇×Q₈	C₂²	1	24	yes	complemented
29	C₂×C₄×C₇×D₄	C₂²	1	24	yes	complemented
30	C₂×C₇×#117	C₂²	1	24	yes	complemented
31	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
35	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
36	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
37	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
38	C₂²×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
39	C₂²×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
40	C₂²×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
41	C₂²×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
42	C₂²×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
43	C₂²×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
44	C₂²×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
45	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
46	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
47	C₂²×C₄²×C₇	C₂²	1	2	yes	—
48	C₂²×#542	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁵×C₇	C₂³	1	1	yes	characteristic
50	C₂⁵×C₇	C₂³	1	2	yes	—
51	C₂²×C₇×Q₈	—	2	8	no	—
52	C₂²×C₇×Q₈	—	4	8	no	—
53	C₂²×C₇×Q₈	C₂³	1	12	yes	—
54	C₂²×C₇×Q₈	C₂³	1	24	yes	—
55	C₂²×C₇×D₄	—	2	8	no	—
56	C₂²×C₇×D₄	—	4	8	no	—
57	C₂²×C₇×D₄	C₂³	1	12	yes	—
58	C₂²×C₇×D₄	C₂³	1	24	yes	—
59	C₂³×C₄×C₇	—	2	8	no	—
60	C₂³×C₄×C₇	—	4	8	no	—
61	C₂³×C₄×C₇	C₂³	1	12	yes	—
62	C₂³×C₄×C₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₇	C₂³	1	2	yes	—
65	C₂³×C₄×C₇	C₂³	1	2	yes	—
66	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₇	C₂³	1	24	yes	—
71	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
72	C₇×C₄⋊Q₈	C₂³	1	64	yes	complemented
73	C₇×#126	C₂³	1	128	yes	complemented
74	C₇×#124	C₂³	1	64	yes	complemented
75	C₇×#124	C₂³	1	128	yes	complemented
76	C₇×#123	C₂³	1	128	yes	complemented
77	C₇×C₂²⋊Q₈	C₂³	1	128	yes	complemented
78	C₇×C₂²⋊Q₈	C₂³	1	64	yes	complemented
79	C₇×#121	C₂³	1	64	yes	—
80	C₄×C₇×Q₈	C₂³	1	64	yes	complemented
81	C₄×C₇×D₄	C₂³	1	64	yes	complemented
82	C₇×#117	C₂³	1	64	yes	complemented
83	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
84	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
85	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
86	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
87	C₂×C₇×C₄⋊C₄	C₂³	1	24	yes	—
88	C₂×C₇×C₄⋊C₄	C₂³	1	24	yes	—
89	C₂×C₇×C₄⋊C₄	C₂³	1	24	yes	—
90	C₂×C₇×C₂²⋊C₄	C₂³	1	12	yes	—
91	C₂×C₇×C₂²⋊C₄	C₂³	1	12	yes	—
92	C₂×C₇×C₂²⋊C₄	C₂³	1	24	yes	—
93	C₂×C₇×C₂²⋊C₄	C₂³	1	24	yes	—
94	C₂×C₇×C₂²⋊C₄	C₂³	1	24	yes	—
95	C₂×C₇×C₂²⋊C₄	C₂³	1	24	yes	—
96	C₂×C₇×C₂²⋊C₄	C₂³	1	24	yes	—
97	C₂×C₄²×C₇	C₂³	1	12	yes	—
98	C₂×C₄²×C₇	C₂³	1	12	yes	—
99	C₂×C₄²×C₇	C₂³	1	24	yes	—
100	C₂×#542	C₂×C₇	1	48	yes	direct product factor
101	C₂²×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
102	C₂²×#126	C₂×C₇	1	2	yes	complemented
103	C₂²×#124	C₂×C₇	1	2	yes	complemented
104	C₂²×#124	C₂×C₇	1	1	yes	characteristic, complemented
105	C₂²×#123	C₂×C₇	1	2	yes	complemented
106	C₂²×C₂²⋊Q₈	C₂×C₇	1	2	yes	complemented
107	C₂²×C₂²⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
108	C₂²×#121	C₂×C₇	1	1	yes	characteristic
109	C₂²×C₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
110	C₂²×C₄×D₄	C₂×C₇	1	1	yes	characteristic, complemented
111	C₂²×#117	C₂×C₇	1	1	yes	characteristic, complemented
112	C₂⁴×C₇	—	2	4	no	—
113	C₂⁴×C₇	—	2	8	no	—
114	C₂⁴×C₇	—	2	8	no	—
115	C₂⁴×C₇	—	2	4	no	—
116	C₂⁴×C₇	—	2	4	no	—
117	C₂⁴×C₇	—	2	8	no	—
118	C₂⁴×C₇	C₂⁴	1	6	yes	—
119	C₂⁴×C₇	C₂⁴	1	12	yes	—
120	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
121	C₂×C₇×Q₈	—	2	48	no	—
122	C₂×C₇×Q₈	—	4	48	no	—
123	C₂×C₇×Q₈	C₂⁴	1	16	yes	—
124	C₂×C₇×Q₈	C₂⁴	1	32	yes	—
125	C₂×C₇×D₄	—	2	48	no	—
126	C₂×C₇×D₄	—	4	48	no	—
127	C₂×C₇×D₄	C₂⁴	1	16	yes	—
128	C₂×C₇×D₄	C₂⁴	1	32	yes	—
129	C₂²×C₄×C₇	—	2	4	no	—
130	C₂²×C₄×C₇	—	2	8	no	—
131	C₂²×C₄×C₇	—	2	4	no	—
132	C₂²×C₄×C₇	—	2	8	no	—
133	C₂²×C₄×C₇	—	2	8	no	—
134	C₂²×C₄×C₇	—	2	4	no	—
135	C₂²×C₄×C₇	—	2	4	no	—
136	C₂²×C₄×C₇	—	2	4	no	—
137	C₂²×C₄×C₇	—	2	48	no	—
138	C₂²×C₄×C₇	—	4	48	no	—
139	C₂²×C₄×C₇	#41	1	8	yes	—
140	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
141	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
142	C₂²×C₄×C₇	C₂⁴	1	12	yes	—
143	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
144	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
145	C₂²×C₄×C₇	C₂⁴	1	12	yes	—
146	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
147	C₂²×C₄×C₇	C₂⁴	1	12	yes	—
148	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
149	C₂²×C₄×C₇	C₂⁴	1	32	yes	—
150	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
151	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
152	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
153	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
154	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
155	C₇×C₄⋊C₄	C₂⁴	1	32	yes	—
156	C₇×C₄⋊C₄	C₂⁴	1	32	yes	—
157	C₇×C₄⋊C₄	C₂⁴	1	32	yes	—
158	C₇×C₂²⋊C₄	C₂⁴	1	16	yes	—
159	C₇×C₂²⋊C₄	C₂⁴	1	16	yes	—
160	C₇×C₂²⋊C₄	C₂⁴	1	32	yes	—
161	C₇×C₂²⋊C₄	C₂⁴	1	32	yes	—
162	C₇×C₂²⋊C₄	C₂⁴	1	32	yes	—
163	C₇×C₂²⋊C₄	C₂⁴	1	32	yes	—
164	C₇×C₂²⋊C₄	C₂⁴	1	32	yes	—
165	C₄²×C₇	C₂⁴	1	16	yes	—
166	C₄²×C₇	C₂⁴	1	16	yes	—
167	C₄²×C₇	C₂⁴	1	32	yes	—
168	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
169	C₂³×Q₈	C₂²×C₇	1	2	yes	—
170	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
171	C₂³×D₄	C₂²×C₇	1	2	yes	—
172	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
173	C₂⁴×C₄	C₂²×C₇	1	2	yes	—
174	#542	C₂²×C₇	1	256	yes	direct product factor
175	C₂×C₄⋊Q₈	C₂²×C₇	1	24	yes	complemented
176	C₂×#126	C₂²×C₇	1	48	yes	complemented
177	C₂×#124	C₂²×C₇	1	24	yes	complemented
178	C₂×#124	C₂²×C₇	1	48	yes	complemented
179	C₂×#123	C₂²×C₇	1	48	yes	complemented
180	C₂×C₂²⋊Q₈	C₂²×C₇	1	48	yes	complemented
181	C₂×C₂²⋊Q₈	C₂²×C₇	1	24	yes	complemented
182	C₂×#121	C₂²×C₇	1	24	yes	—
183	C₂×C₄×Q₈	C₂²×C₇	1	24	yes	complemented
184	C₂×C₄×D₄	C₂²×C₇	1	24	yes	complemented
185	C₂×#117	C₂²×C₇	1	24	yes	complemented
186	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
187	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
188	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
189	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
190	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
191	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
192	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
193	C₂²×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
194	C₂²×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
195	C₂²×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
196	C₂²×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
197	C₂²×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
198	C₂²×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
199	C₂²×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
200	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
201	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
202	C₂²×C₄²	C₂²×C₇	1	2	yes	—
203	C₂³×C₇	—	2	12	no	—
204	C₂³×C₇	—	2	24	no	—
205	C₂³×C₇	—	2	24	no	—
206	C₂³×C₇	—	2	12	no	—
207	C₂³×C₇	—	2	12	no	—
208	C₂³×C₇	—	2	24	no	—
209	C₂³×C₇	—	4	16	no	—
210	C₂³×C₇	—	4	32	no	—
211	C₂³×C₇	C₂×#41	1	4	yes	—
212	C₂³×C₇	H(5,𝔽₂)	1	4	yes	—
213	C₂³×C₇	2₋¹⁺⁴	1	4	yes	—
214	C₂³×C₇	C₂⁵	1	4	yes	—
215	C₂³×C₇	C₂⁵	1	8	yes	—
216	C₂³×C₇	C₂⁵	1	3	yes	—
217	C₇×Q₈	—	2	32	no	—
218	C₇×Q₈	—	4	32	no	—
219	C₇×D₄	—	2	32	no	—
220	C₇×D₄	—	4	32	no	—
221	C₂×C₄×C₇	—	2	12	no	—
222	C₂×C₄×C₇	—	2	24	no	—
223	C₂×C₄×C₇	—	2	12	no	—
224	C₂×C₄×C₇	—	2	24	no	—
225	C₂×C₄×C₇	—	2	24	no	—
226	C₂×C₄×C₇	—	2	12	no	—
227	C₂×C₄×C₇	—	2	12	no	—
228	C₂×C₄×C₇	—	2	12	no	—
229	C₂×C₄×C₇	—	2	32	no	—
230	C₂×C₄×C₇	—	4	32	no	—
231	C₂×C₄×C₇	C₂×#41	1	24	yes	—
232	C₂×C₄×C₇	C₂⁵	1	4	yes	—
233	C₂×C₄×C₇	C₂⁵	1	4	yes	—
234	C₂×C₄×C₇	C₂⁵	1	8	yes	—
235	C₂×C₄×C₇	C₂⁵	1	4	yes	—
236	C₂×C₄×C₇	C₂⁵	1	4	yes	—
237	C₂×C₄×C₇	C₂⁵	1	8	yes	—
238	C₂×C₄×C₇	C₂⁵	1	4	yes	—
239	C₂×C₄×C₇	C₂⁵	1	8	yes	—
240	C₂×C₄×C₇	C₂⁵	1	4	yes	—
241	C₂⁵	C₂³×C₇	1	1	yes	characteristic
242	C₂⁵	C₂³×C₇	1	2	yes	—
243	C₂²×Q₈	—	2	8	no	—
244	C₂²×Q₈	—	4	8	no	—
245	C₂²×Q₈	C₂³×C₇	1	12	yes	—
246	C₂²×Q₈	C₂³×C₇	1	24	yes	—
247	C₂²×D₄	—	2	8	no	—
248	C₂²×D₄	—	4	8	no	—
249	C₂²×D₄	C₂³×C₇	1	12	yes	—
250	C₂²×D₄	C₂³×C₇	1	24	yes	—
251	C₂³×C₄	—	2	8	no	—
252	C₂³×C₄	—	4	8	no	—
253	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
254	C₂³×C₄	C₂³×C₇	1	2	yes	—
255	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
256	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
257	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
258	C₂³×C₄	C₂³×C₇	1	2	yes	—
259	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
260	C₂³×C₄	C₂³×C₇	1	2	yes	—
261	C₂³×C₄	C₂³×C₇	1	12	yes	—
262	C₂³×C₄	C₂³×C₇	1	24	yes	—
263	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
264	C₄⋊Q₈	C₂³×C₇	1	64	yes	complemented
265	#126	C₂³×C₇	1	128	yes	complemented
266	#124	C₂³×C₇	1	64	yes	complemented
267	#124	C₂³×C₇	1	128	yes	complemented
268	#123	C₂³×C₇	1	128	yes	complemented
269	C₂²⋊Q₈	C₂³×C₇	1	128	yes	complemented
270	C₂²⋊Q₈	C₂³×C₇	1	64	yes	complemented
271	#121	C₂³×C₇	1	64	yes	—
272	C₄×Q₈	C₂³×C₇	1	64	yes	complemented
273	C₄×D₄	C₂³×C₇	1	64	yes	complemented
274	#117	C₂³×C₇	1	64	yes	complemented
275	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
276	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
277	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
278	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
279	C₂×C₄⋊C₄	C₂³×C₇	1	24	yes	—
280	C₂×C₄⋊C₄	C₂³×C₇	1	24	yes	—
281	C₂×C₄⋊C₄	C₂³×C₇	1	24	yes	—
282	C₂×C₂²⋊C₄	C₂³×C₇	1	12	yes	—
283	C₂×C₂²⋊C₄	C₂³×C₇	1	12	yes	—
284	C₂×C₂²⋊C₄	C₂³×C₇	1	24	yes	—
285	C₂×C₂²⋊C₄	C₂³×C₇	1	24	yes	—
286	C₂×C₂²⋊C₄	C₂³×C₇	1	24	yes	—
287	C₂×C₂²⋊C₄	C₂³×C₇	1	24	yes	—
288	C₂×C₂²⋊C₄	C₂³×C₇	1	24	yes	—
289	C₂×C₄²	C₂³×C₇	1	12	yes	—
290	C₂×C₄²	C₂³×C₇	1	12	yes	—
291	C₂×C₄²	C₂³×C₇	1	24	yes	—
292	C₂²×C₇	—	2	4	no	—
293	C₂²×C₇	—	2	8	no	—
294	C₂²×C₇	—	2	8	no	—
295	C₂²×C₇	—	2	4	no	—
296	C₂²×C₇	—	2	4	no	—
297	C₂²×C₇	—	2	8	no	—
298	C₂²×C₇	—	4	24	no	—
299	C₂²×C₇	—	4	48	no	—
300	C₂²×C₇	#542	1	16	yes	direct product factor
301	C₂²×C₇	C₂²×#41	1	6	yes	—
302	C₂²×C₇	C₂×H(5,𝔽₂)	1	6	yes	—
303	C₂²×C₇	C₂×2₋¹⁺⁴	1	6	yes	—
304	C₂²×C₇	C₂⁶	1	1	yes	characteristic
305	C₄×C₇	—	2	4	no	—
306	C₄×C₇	—	2	8	no	—
307	C₄×C₇	—	2	4	no	—
308	C₄×C₇	—	2	8	no	—
309	C₄×C₇	—	2	8	no	—
310	C₄×C₇	—	2	4	no	—
311	C₄×C₇	—	2	4	no	—
312	C₄×C₇	—	2	4	no	—
313	C₄×C₇	C₂²×#41	1	8	yes	—
314	C₂⁴	—	2	4	no	—
315	C₂⁴	—	2	8	no	—
316	C₂⁴	—	2	8	no	—
317	C₂⁴	—	2	4	no	—
318	C₂⁴	—	2	4	no	—
319	C₂⁴	—	2	8	no	—
320	C₂⁴	C₂⁴×C₇	1	6	yes	—
321	C₂⁴	C₂⁴×C₇	1	12	yes	—
322	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
323	C₂×Q₈	—	2	48	no	—
324	C₂×Q₈	—	4	48	no	—
325	C₂×Q₈	C₂⁴×C₇	1	16	yes	—
326	C₂×Q₈	C₂⁴×C₇	1	32	yes	—
327	C₂×D₄	—	2	48	no	—
328	C₂×D₄	—	4	48	no	—
329	C₂×D₄	C₂⁴×C₇	1	16	yes	—
330	C₂×D₄	C₂⁴×C₇	1	32	yes	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	8	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	8	no	—
335	C₂²×C₄	—	2	8	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	4	no	—
339	C₂²×C₄	—	2	48	no	—
340	C₂²×C₄	—	4	48	no	—
341	C₂²×C₄	C₇×#41	1	8	yes	—
342	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
343	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
344	C₂²×C₄	C₂⁴×C₇	1	12	yes	—
345	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
346	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
347	C₂²×C₄	C₂⁴×C₇	1	12	yes	—
348	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
349	C₂²×C₄	C₂⁴×C₇	1	12	yes	—
350	C₂²×C₄	C₂⁴×C₇	1	16	yes	—
351	C₂²×C₄	C₂⁴×C₇	1	32	yes	—
352	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
353	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
354	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
355	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
356	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
357	C₄⋊C₄	C₂⁴×C₇	1	32	yes	—
358	C₄⋊C₄	C₂⁴×C₇	1	32	yes	—
359	C₄⋊C₄	C₂⁴×C₇	1	32	yes	—
360	C₂²⋊C₄	C₂⁴×C₇	1	16	yes	—
361	C₂²⋊C₄	C₂⁴×C₇	1	16	yes	—
362	C₂²⋊C₄	C₂⁴×C₇	1	32	yes	—
363	C₂²⋊C₄	C₂⁴×C₇	1	32	yes	—
364	C₂²⋊C₄	C₂⁴×C₇	1	32	yes	—
365	C₂²⋊C₄	C₂⁴×C₇	1	32	yes	—
366	C₂²⋊C₄	C₂⁴×C₇	1	32	yes	—
367	C₄²	C₂⁴×C₇	1	16	yes	—
368	C₄²	C₂⁴×C₇	1	16	yes	—
369	C₄²	C₂⁴×C₇	1	32	yes	—
370	C₂×C₇	—	4	4	no	—
371	C₂×C₇	—	4	8	no	—
372	C₂×C₇	C₂×#542	1	12	yes	direct product factor
373	C₂×C₇	C₂³×#41	1	1	yes	characteristic
374	C₂×C₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
375	C₂×C₇	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
376	C₂³	—	2	12	no	—
377	C₂³	—	2	24	no	—
378	C₂³	—	2	24	no	—
379	C₂³	—	2	12	no	—
380	C₂³	—	2	12	no	—
381	C₂³	—	2	24	no	—
382	C₂³	—	4	16	no	—
383	C₂³	—	4	32	no	—
384	C₂³	C₂×C₇×#41	1	4	yes	—
385	C₂³	C₇×H(5,𝔽₂)	1	4	yes	—
386	C₂³	C₇×2₋¹⁺⁴	1	4	yes	—
387	C₂³	C₂⁵×C₇	1	4	yes	—
388	C₂³	C₂⁵×C₇	1	8	yes	—
389	C₂³	C₂⁵×C₇	1	3	yes	—
390	Q₈	—	2	32	no	—
391	Q₈	—	4	32	no	—
392	D₄	—	2	32	no	—
393	D₄	—	4	32	no	—
394	C₂×C₄	—	2	12	no	—
395	C₂×C₄	—	2	24	no	—
396	C₂×C₄	—	2	12	no	—
397	C₂×C₄	—	2	24	no	—
398	C₂×C₄	—	2	24	no	—
399	C₂×C₄	—	2	12	no	—
400	C₂×C₄	—	2	12	no	—
401	C₂×C₄	—	2	12	no	—
402	C₂×C₄	—	2	32	no	—
403	C₂×C₄	—	4	32	no	—
404	C₂×C₄	C₂×C₇×#41	1	24	yes	—
405	C₂×C₄	C₂⁵×C₇	1	4	yes	—
406	C₂×C₄	C₂⁵×C₇	1	4	yes	—
407	C₂×C₄	C₂⁵×C₇	1	8	yes	—
408	C₂×C₄	C₂⁵×C₇	1	4	yes	—
409	C₂×C₄	C₂⁵×C₇	1	4	yes	—
410	C₂×C₄	C₂⁵×C₇	1	8	yes	—
411	C₂×C₄	C₂⁵×C₇	1	4	yes	—
412	C₂×C₄	C₂⁵×C₇	1	8	yes	—
413	C₂×C₄	C₂⁵×C₇	1	4	yes	—
414	C₇	C₂²×#542	1	1	yes	characteristic, direct product factor, Sylow
415	C₂²	—	2	4	no	—
416	C₂²	—	2	8	no	—
417	C₂²	—	2	8	no	—
418	C₂²	—	2	4	no	—
419	C₂²	—	2	4	no	—
420	C₂²	—	2	8	no	—
421	C₂²	—	4	24	no	—
422	C₂²	—	4	48	no	—
423	C₂²	C₇×#542	1	16	yes	direct product factor
424	C₂²	C₂²×C₇×#41	1	6	yes	—
425	C₂²	C₂×C₇×H(5,𝔽₂)	1	6	yes	—
426	C₂²	C₂×C₇×2₋¹⁺⁴	1	6	yes	—
427	C₂²	C₂⁶×C₇	1	1	yes	characteristic
428	C₄	—	2	4	no	—
429	C₄	—	2	8	no	—
430	C₄	—	2	4	no	—
431	C₄	—	2	8	no	—
432	C₄	—	2	8	no	—
433	C₄	—	2	4	no	—
434	C₄	—	2	4	no	—
435	C₄	—	2	4	no	—
436	C₄	C₂²×C₇×#41	1	8	yes	—
437	C₂	—	4	4	no	—
438	C₂	—	4	8	no	—
439	C₂	C₂×C₇×#542	1	12	yes	direct product factor
440	C₂	C₂³×C₇×#41	1	1	yes	characteristic
441	C₂	C₂²×C₇×H(5,𝔽₂)	1	1	yes	characteristic
442	C₂	C₂²×C₇×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₇
prime radical	C₂²×C₇×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#542	C₂	direct (N × H)
2. C₂²×C₇×C₄⋊Q₈	C₂	semidirect
3. C₂²×C₇×#126	C₂	semidirect
4. C₂²×C₇×#124	C₂	semidirect
5. C₂²×C₇×#124	C₂	semidirect
6. C₂²×C₇×#123	C₂	semidirect
7. C₂²×C₇×C₂²⋊Q₈	C₂	semidirect
8. C₂²×C₇×C₂²⋊Q₈	C₂	semidirect
10. C₂²×C₄×C₇×Q₈	C₂	semidirect
11. C₂²×C₄×C₇×D₄	C₂	semidirect
12. C₂²×C₇×#117	C₂	semidirect
19. C₇×#542	C₂²	direct (N × H)
20. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
21. C₂×C₇×#126	C₂²	semidirect
22. C₂×C₇×#124	C₂²	semidirect
23. C₂×C₇×#124	C₂²	semidirect
24. C₂×C₇×#123	C₂²	semidirect
25. C₂×C₇×C₂²⋊Q₈	C₂²	semidirect
26. C₂×C₇×C₂²⋊Q₈	C₂²	semidirect
28. C₂×C₄×C₇×Q₈	C₂²	semidirect
29. C₂×C₄×C₇×D₄	C₂²	semidirect
30. C₂×C₇×#117	C₂²	semidirect
48. C₂²×#542	C₇	direct (N × H)
72. C₇×C₄⋊Q₈	C₂³	semidirect
73. C₇×#126	C₂³	semidirect
74. C₇×#124	C₂³	semidirect
75. C₇×#124	C₂³	semidirect
76. C₇×#123	C₂³	semidirect
77. C₇×C₂²⋊Q₈	C₂³	semidirect
78. C₇×C₂²⋊Q₈	C₂³	semidirect
80. C₄×C₇×Q₈	C₂³	semidirect
81. C₄×C₇×D₄	C₂³	semidirect
82. C₇×#117	C₂³	semidirect
100. C₂×#542	C₂×C₇	direct (N × H)
101. C₂²×C₄⋊Q₈	C₂×C₇	semidirect
102. C₂²×#126	C₂×C₇	semidirect
103. C₂²×#124	C₂×C₇	semidirect
104. C₂²×#124	C₂×C₇	semidirect
105. C₂²×#123	C₂×C₇	semidirect
106. C₂²×C₂²⋊Q₈	C₂×C₇	semidirect
107. C₂²×C₂²⋊Q₈	C₂×C₇	semidirect
109. C₂²×C₄×Q₈	C₂×C₇	semidirect
110. C₂²×C₄×D₄	C₂×C₇	semidirect
111. C₂²×#117	C₂×C₇	semidirect
174. #542	C₂²×C₇	direct (N × H)
175. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
176. C₂×#126	C₂²×C₇	semidirect
177. C₂×#124	C₂²×C₇	semidirect
178. C₂×#124	C₂²×C₇	semidirect
179. C₂×#123	C₂²×C₇	semidirect
180. C₂×C₂²⋊Q₈	C₂²×C₇	semidirect
181. C₂×C₂²⋊Q₈	C₂²×C₇	semidirect
183. C₂×C₄×Q₈	C₂²×C₇	semidirect
184. C₂×C₄×D₄	C₂²×C₇	semidirect
185. C₂×#117	C₂²×C₇	semidirect
264. C₄⋊Q₈	C₂³×C₇	semidirect
265. #126	C₂³×C₇	semidirect
266. #124	C₂³×C₇	semidirect
267. #124	C₂³×C₇	semidirect
268. #123	C₂³×C₇	semidirect
269. C₂²⋊Q₈	C₂³×C₇	semidirect
270. C₂²⋊Q₈	C₂³×C₇	semidirect
272. C₄×Q₈	C₂³×C₇	semidirect
273. C₄×D₄	C₂³×C₇	semidirect
274. #117	C₂³×C₇	semidirect
300. C₂²×C₇	#542	direct (N × H)
372. C₂×C₇	C₂×#542	direct (N × H)
414. C₇	C₂²×#542	direct (N × H)
423. C₂²	C₇×#542	direct (N × H)
439. C₂	C₂×C₇×#542	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×#542	2
C₂²×C₇×C₄⋊Q₈	2
C₂²×C₇×#126	2
C₂²×C₇×#124	2
C₂²×C₇×#123	2
C₂²×C₇×C₂²⋊Q₈	2
C₂²×C₇×#121	2
C₂²×C₄×C₇×Q₈	2
C₂²×C₄×C₇×D₄	2
C₂²×C₇×#117	2
C₂²×#542	7

Subgroups — C22×C7×#542 #13239894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#542 #13239894