Subgroups — C₂×C₄₇×#2354 #26073815

Order 12032.

3678 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#117×C₄₇	C₂	1	2	yes	maximal, complemented
3	C₄₇×#2354	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄₇×#390	C₂	1	2	yes	maximal
5	C₂×C₄₇×#388	C₂	1	4	yes	maximal, complemented
6	C₂×C₄₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#380	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×C₄⋊C₄×C₄₇	C₂	1	2	yes	maximal, complemented
10	C₂³×D₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₄₇	C₄	1	16	yes	complemented
13	C₂×C₄×D₄×C₄₇	C₂²	1	8	yes	—
14	C₂×#117×C₄₇	—	2	16	no	—
15	C₂×#117×C₄₇	C₂²	1	16	yes	complemented
16	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
23	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
24	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
27	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#390	C₂²	1	16	yes	—
29	C₄₇×#388	C₂²	1	32	yes	complemented
30	C₄₇×#384	C₂²	1	8	yes	complemented
31	C₄₇×#380	C₂²	1	16	yes	complemented
32	C₄×C₄⋊C₄×C₄₇	C₂²	1	16	yes	complemented
33	C₄×C₄⋊C₄×C₄₇	C₂²	1	8	yes	complemented
34	C₂×#95×C₄₇	C₂²	1	4	yes	—
35	C₂×#95×C₄₇	C₂²	1	2	yes	—
36	C₂⁵×C₄₇	C₂³	1	2	yes	—
37	C₂²×D₄×C₄₇	—	2	4	no	—
38	C₂²×D₄×C₄₇	—	2	4	no	—
39	C₂²×D₄×C₄₇	C₂×C₄	1	8	yes	—
40	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
41	C₂³×C₄×C₄₇	—	2	8	no	—
42	C₂³×C₄×C₄₇	—	2	8	no	—
43	C₂³×C₄×C₄₇	C₂×C₄	1	16	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
45	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	8	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₄×D₄×C₄₇	C₂×C₄	1	64	yes	complemented
53	#117×C₄₇	—	2	64	no	—
54	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
55	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
56	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
57	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
58	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
59	C₂×C₄⋊C₄×C₄₇	D₄	1	8	yes	complemented
60	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
61	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
63	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
64	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
65	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
66	C₂×C₂²⋊C₄×C₄₇	—	2	16	no	—
67	C₂×C₂²⋊C₄×C₄₇	—	4	8	no	—
68	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
69	C₂×C₂²⋊C₄×C₄₇	C₂³	1	16	yes	—
70	C₂×C₂²⋊C₄×C₄₇	C₂³	1	16	yes	—
71	C₂×C₂²⋊C₄×C₄₇	C₂³	1	8	yes	—
72	C₂×C₄²×C₄₇	—	2	16	no	—
73	C₂×C₄²×C₄₇	—	2	8	no	—
74	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
75	C₂×C₄²×C₄₇	C₂³	1	16	yes	—
76	C₂×C₄²×C₄₇	C₂³	1	8	yes	—
77	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
78	#95×C₄₇	C₂³	1	16	yes	—
79	#95×C₄₇	C₂³	1	8	yes	—
80	C₂⁴×C₄₇	—	2	4	no	—
81	C₂⁴×C₄₇	—	2	4	no	—
82	C₂⁴×C₄₇	—	2	4	no	—
83	C₂⁴×C₄₇	—	2	4	no	—
84	C₂⁴×C₄₇	—	2	4	no	—
85	C₂⁴×C₄₇	—	2	4	no	—
86	C₂⁴×C₄₇	C₂²×C₄	1	8	yes	—
87	C₂⁴×C₄₇	C₂⁴	1	4	yes	—
88	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₂×D₄×C₄₇	—	2	32	no	—
90	C₂×D₄×C₄₇	—	2	8	no	—
91	C₂×D₄×C₄₇	—	2	8	no	—
92	C₂×D₄×C₄₇	C₂²×C₄	1	16	yes	—
93	C₂²×C₄×C₄₇	—	2	4	no	—
94	C₂²×C₄×C₄₇	—	2	4	no	—
95	C₂²×C₄×C₄₇	—	2	4	no	—
96	C₂²×C₄×C₄₇	—	2	2	no	—
97	C₂²×C₄×C₄₇	—	2	2	no	—
98	C₂²×C₄×C₄₇	—	2	4	no	—
99	C₂²×C₄×C₄₇	—	2	64	no	—
100	C₂²×C₄×C₄₇	—	2	16	no	—
101	C₂²×C₄×C₄₇	—	2	16	no	—
102	C₂²×C₄×C₄₇	—	2	8	no	—
103	C₂²×C₄×C₄₇	—	2	4	no	—
104	C₂²×C₄×C₄₇	—	2	8	no	—
105	C₂²×C₄×C₄₇	—	2	4	no	—
106	C₂²×C₄×C₄₇	—	2	4	no	—
107	C₂²×C₄×C₄₇	—	2	8	no	—
108	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
109	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
110	C₂²×C₄×C₄₇	C₂²×C₄	1	32	yes	—
111	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
112	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
113	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
114	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
115	C₂²×C₄×C₄₇	#41	1	4	yes	—
116	C₂²×C₄×C₄₇	#41	1	4	yes	—
117	C₂²×C₄×C₄₇	#41	1	4	yes	—
118	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
119	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
121	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
122	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
123	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
124	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
125	C₄⋊C₄×C₄₇	—	2	16	no	—
126	C₄⋊C₄×C₄₇	—	2	16	no	—
127	C₄⋊C₄×C₄₇	—	2	8	no	—
128	C₄⋊C₄×C₄₇	—	4	8	no	—
129	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
130	C₄⋊C₄×C₄₇	C₂×D₄	1	16	yes	complemented
131	C₂²⋊C₄×C₄₇	—	2	32	no	—
132	C₂²⋊C₄×C₄₇	—	4	16	no	—
133	C₂²⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
134	C₄²×C₄₇	—	2	32	no	—
135	C₄²×C₄₇	—	2	16	no	—
136	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
137	C₂³×C₄₇	—	2	16	no	—
138	C₂³×C₄₇	—	2	16	no	—
139	C₂³×C₄₇	—	2	16	no	—
140	C₂³×C₄₇	—	2	4	no	—
141	C₂³×C₄₇	—	2	4	no	—
142	C₂³×C₄₇	—	2	4	no	—
143	C₂³×C₄₇	—	2	4	no	—
144	C₂³×C₄₇	—	2	4	no	—
145	C₂³×C₄₇	—	2	4	no	—
146	C₂³×C₄₇	—	4	8	no	—
147	C₂³×C₄₇	—	4	8	no	—
148	C₂³×C₄₇	—	4	8	no	—
149	C₂³×C₄₇	—	4	8	no	—
150	C₂³×C₄₇	C₂³×C₄	1	8	yes	—
151	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
152	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
153	C₂³×C₄₇	C₂×#41	1	2	yes	—
154	C₂³×C₄₇	C₂×#41	1	2	yes	—
155	C₂³×C₄₇	C₂×#41	1	2	yes	—
156	C₂³×C₄₇	H(5,𝔽₂)	1	2	yes	—
157	C₂³×C₄₇	2₋¹⁺⁴	1	2	yes	—
158	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
159	D₄×C₄₇	—	2	32	no	—
160	C₂×C₄×C₄₇	—	2	16	no	—
161	C₂×C₄×C₄₇	—	2	8	no	—
162	C₂×C₄×C₄₇	—	2	16	no	—
163	C₂×C₄×C₄₇	—	2	4	no	—
164	C₂×C₄×C₄₇	—	2	4	no	—
165	C₂×C₄×C₄₇	—	2	4	no	—
166	C₂×C₄×C₄₇	—	2	2	no	—
167	C₂×C₄×C₄₇	—	2	2	no	—
168	C₂×C₄×C₄₇	—	2	4	no	—
169	C₂×C₄×C₄₇	—	2	64	no	—
170	C₂×C₄×C₄₇	—	2	8	no	—
171	C₂×C₄×C₄₇	—	2	4	no	—
172	C₂×C₄×C₄₇	—	2	8	no	—
173	C₂×C₄×C₄₇	—	2	4	no	—
174	C₂×C₄×C₄₇	—	2	4	no	—
175	C₂×C₄×C₄₇	—	2	8	no	—
176	C₂×C₄×C₄₇	—	4	16	no	—
177	C₂×C₄×C₄₇	—	4	8	no	—
178	C₂×C₄×C₄₇	#117	1	16	yes	complemented
179	C₂×C₄×C₄₇	C₄×D₄	1	16	yes	—
180	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
181	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
182	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
183	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
184	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
185	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
186	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
187	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
188	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
189	C₂×#2354	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂²×C₄₇	—	2	8	no	—
191	C₂²×C₄₇	—	2	8	no	—
192	C₂²×C₄₇	—	2	8	no	—
193	C₂²×C₄₇	—	4	32	no	—
194	C₂²×C₄₇	—	4	4	no	—
195	C₂²×C₄₇	—	4	4	no	—
196	C₂²×C₄₇	—	4	4	no	—
197	C₂²×C₄₇	—	4	4	no	—
198	C₂²×C₄₇	C₂×#117	1	4	yes	—
199	C₂²×C₄₇	C₂×C₄×D₄	1	4	yes	—
200	C₂²×C₄₇	#520	1	4	yes	—
201	C₂²×C₄₇	#547	1	4	yes	—
202	C₂²×C₄₇	#550	1	4	yes	—
203	C₂²×C₄₇	#555	1	4	yes	—
204	C₂²×C₄₇	#556	1	4	yes	—
205	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
206	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
207	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
208	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
209	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
210	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
211	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
212	C₄×C₄₇	—	2	8	no	—
213	C₄×C₄₇	—	2	4	no	—
214	C₄×C₄₇	—	2	8	no	—
215	C₄×C₄₇	—	4	8	no	—
216	C₄×C₄₇	—	4	4	no	—
217	C₄×C₄₇	C₂×#117	1	8	yes	complemented
218	C₄×C₄₇	C₂×C₄×D₄	1	8	yes	—
219	C₂²×C₄×D₄	C₂×C₄₇	1	1	yes	characteristic
220	C₂²×#117	C₂×C₄₇	1	2	yes	complemented
221	#2354	C₂×C₄₇	1	16	yes	direct product factor
222	C₂×#390	C₂×C₄₇	1	2	yes	—
223	C₂×#388	C₂×C₄₇	1	4	yes	complemented
224	C₂×#384	C₂×C₄₇	1	1	yes	characteristic, complemented
225	C₂×#380	C₂×C₄₇	1	2	yes	complemented
226	C₂×C₄×C₄⋊C₄	C₂×C₄₇	1	1	yes	characteristic, complemented
227	C₂×C₄×C₄⋊C₄	C₂×C₄₇	1	2	yes	complemented
228	C₂×C₄₇	—	4	8	no	—
229	C₂×C₄₇	#2354	1	8	yes	direct product factor
230	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
231	C₂×C₄₇	C₂²×C₄×D₄	1	1	yes	characteristic
232	C₂×C₄₇	C₂×#520	1	1	yes	characteristic
233	C₂×C₄₇	C₂×#547	1	1	yes	characteristic
234	C₂×C₄₇	C₂×#550	1	1	yes	characteristic
235	C₂×C₄₇	C₂×#555	1	1	yes	characteristic
236	C₂×C₄₇	C₂×#556	1	1	yes	characteristic
237	C₂³×D₄	C₂²×C₄₇	1	1	yes	characteristic
238	C₂⁴×C₄	C₂²×C₄₇	1	2	yes	—
239	C₂×C₄×D₄	C₄×C₄₇	1	16	yes	complemented
240	C₂×C₄×D₄	C₂²×C₄₇	1	8	yes	—
241	C₂×#117	—	2	16	no	—
242	C₂×#117	C₂²×C₄₇	1	16	yes	complemented
243	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
244	C₂²×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
245	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
246	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
247	C₂²×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
248	C₂²×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
249	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	4	yes	—
250	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	4	yes	—
251	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	2	yes	—
252	C₂²×C₄²	C₂²×C₄₇	1	4	yes	—
253	C₂²×C₄²	C₂²×C₄₇	1	2	yes	—
254	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
255	#390	C₂²×C₄₇	1	16	yes	—
256	#388	C₂²×C₄₇	1	32	yes	complemented
257	#384	C₂²×C₄₇	1	8	yes	complemented
258	#380	C₂²×C₄₇	1	16	yes	complemented
259	C₄×C₄⋊C₄	C₂²×C₄₇	1	16	yes	complemented
260	C₄×C₄⋊C₄	C₂²×C₄₇	1	8	yes	complemented
261	C₂×#95	C₂²×C₄₇	1	4	yes	—
262	C₂×#95	C₂²×C₄₇	1	2	yes	—
263	C₄₇	C₂×#2354	1	1	yes	characteristic, direct product factor, Sylow
264	C₂⁵	C₂³×C₄₇	1	2	yes	—
265	C₂²×D₄	—	2	4	no	—
266	C₂²×D₄	—	2	4	no	—
267	C₂²×D₄	C₂×C₄×C₄₇	1	8	yes	—
268	C₂²×D₄	C₂³×C₄₇	1	4	yes	—
269	C₂³×C₄	—	2	8	no	—
270	C₂³×C₄	—	2	8	no	—
271	C₂³×C₄	C₂×C₄×C₄₇	1	16	yes	—
272	C₂³×C₄	C₂³×C₄₇	1	4	yes	—
273	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
274	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
275	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
276	C₂³×C₄	C₂³×C₄₇	1	8	yes	—
277	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
278	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
279	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
280	C₄×D₄	C₂×C₄×C₄₇	1	64	yes	complemented
281	#117	—	2	64	no	—
282	C₂×C₄⋊C₄	—	2	8	no	—
283	C₂×C₄⋊C₄	—	2	8	no	—
284	C₂×C₄⋊C₄	—	2	4	no	—
285	C₂×C₄⋊C₄	—	4	4	no	—
286	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	8	yes	—
287	C₂×C₄⋊C₄	D₄×C₄₇	1	8	yes	complemented
288	C₂×C₄⋊C₄	C₂³×C₄₇	1	8	yes	—
289	C₂×C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
290	C₂×C₄⋊C₄	C₂³×C₄₇	1	8	yes	—
291	C₂×C₄⋊C₄	C₂³×C₄₇	1	8	yes	—
292	C₂×C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
293	C₂×C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
294	C₂×C₂²⋊C₄	—	2	16	no	—
295	C₂×C₂²⋊C₄	—	4	8	no	—
296	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	16	yes	—
297	C₂×C₂²⋊C₄	C₂³×C₄₇	1	16	yes	—
298	C₂×C₂²⋊C₄	C₂³×C₄₇	1	16	yes	—
299	C₂×C₂²⋊C₄	C₂³×C₄₇	1	8	yes	—
300	C₂×C₄²	—	2	16	no	—
301	C₂×C₄²	—	2	8	no	—
302	C₂×C₄²	C₂×C₄×C₄₇	1	8	yes	—
303	C₂×C₄²	C₂³×C₄₇	1	16	yes	—
304	C₂×C₄²	C₂³×C₄₇	1	8	yes	—
305	C₂×C₄²	C₂³×C₄₇	1	4	yes	—
306	#95	C₂³×C₄₇	1	16	yes	—
307	#95	C₂³×C₄₇	1	8	yes	—
308	C₂⁴	—	2	4	no	—
309	C₂⁴	—	2	4	no	—
310	C₂⁴	—	2	4	no	—
311	C₂⁴	—	2	4	no	—
312	C₂⁴	—	2	4	no	—
313	C₂⁴	—	2	4	no	—
314	C₂⁴	C₂²×C₄×C₄₇	1	8	yes	—
315	C₂⁴	C₂⁴×C₄₇	1	4	yes	—
316	C₂⁴	C₂⁴×C₄₇	1	1	yes	characteristic
317	C₂×D₄	—	2	32	no	—
318	C₂×D₄	—	2	8	no	—
319	C₂×D₄	—	2	8	no	—
320	C₂×D₄	C₂²×C₄×C₄₇	1	16	yes	—
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	2	4	no	—
323	C₂²×C₄	—	2	4	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	64	no	—
328	C₂²×C₄	—	2	16	no	—
329	C₂²×C₄	—	2	16	no	—
330	C₂²×C₄	—	2	8	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	8	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	8	no	—
336	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
337	C₂²×C₄	C₂²×C₄×C₄₇	1	8	yes	—
338	C₂²×C₄	C₂²×C₄×C₄₇	1	32	yes	—
339	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
340	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
341	C₂²×C₄	C₂×D₄×C₄₇	1	8	yes	—
342	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
343	C₂²×C₄	#41×C₄₇	1	4	yes	—
344	C₂²×C₄	#41×C₄₇	1	4	yes	—
345	C₂²×C₄	#41×C₄₇	1	4	yes	—
346	C₂²×C₄	C₂⁴×C₄₇	1	4	yes	—
347	C₂²×C₄	C₂⁴×C₄₇	1	8	yes	—
348	C₂²×C₄	C₂⁴×C₄₇	1	4	yes	—
349	C₂²×C₄	C₂⁴×C₄₇	1	2	yes	—
350	C₂²×C₄	C₂⁴×C₄₇	1	4	yes	—
351	C₂²×C₄	C₂⁴×C₄₇	1	2	yes	—
352	C₂²×C₄	C₂⁴×C₄₇	1	2	yes	—
353	C₄⋊C₄	—	2	16	no	—
354	C₄⋊C₄	—	2	16	no	—
355	C₄⋊C₄	—	2	8	no	—
356	C₄⋊C₄	—	4	8	no	—
357	C₄⋊C₄	C₂²×C₄×C₄₇	1	16	yes	—
358	C₄⋊C₄	C₂×D₄×C₄₇	1	16	yes	complemented
359	C₂²⋊C₄	—	2	32	no	—
360	C₂²⋊C₄	—	4	16	no	—
361	C₂²⋊C₄	C₂²×C₄×C₄₇	1	32	yes	—
362	C₄²	—	2	32	no	—
363	C₄²	—	2	16	no	—
364	C₄²	C₂²×C₄×C₄₇	1	16	yes	—
365	C₂³	—	2	16	no	—
366	C₂³	—	2	16	no	—
367	C₂³	—	2	16	no	—
368	C₂³	—	2	4	no	—
369	C₂³	—	2	4	no	—
370	C₂³	—	2	4	no	—
371	C₂³	—	2	4	no	—
372	C₂³	—	2	4	no	—
373	C₂³	—	2	4	no	—
374	C₂³	—	4	8	no	—
375	C₂³	—	4	8	no	—
376	C₂³	—	4	8	no	—
377	C₂³	—	4	8	no	—
378	C₂³	C₂³×C₄×C₄₇	1	8	yes	—
379	C₂³	C₂³×C₄×C₄₇	1	2	yes	—
380	C₂³	C₂²×D₄×C₄₇	1	2	yes	—
381	C₂³	C₂×#41×C₄₇	1	2	yes	—
382	C₂³	C₂×#41×C₄₇	1	2	yes	—
383	C₂³	C₂×#41×C₄₇	1	2	yes	—
384	C₂³	H(5,𝔽₂)×C₄₇	1	2	yes	—
385	C₂³	2₋¹⁺⁴×C₄₇	1	2	yes	—
386	C₂³	C₂⁵×C₄₇	1	1	yes	characteristic
387	D₄	—	2	32	no	—
388	C₂×C₄	—	2	16	no	—
389	C₂×C₄	—	2	8	no	—
390	C₂×C₄	—	2	16	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	64	no	—
398	C₂×C₄	—	2	8	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	8	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	8	no	—
404	C₂×C₄	—	4	16	no	—
405	C₂×C₄	—	4	8	no	—
406	C₂×C₄	#117×C₄₇	1	16	yes	complemented
407	C₂×C₄	C₄×D₄×C₄₇	1	16	yes	—
408	C₂×C₄	C₂³×C₄×C₄₇	1	4	yes	—
409	C₂×C₄	C₂³×C₄×C₄₇	1	8	yes	—
410	C₂×C₄	C₂³×C₄×C₄₇	1	4	yes	—
411	C₂×C₄	C₂³×C₄×C₄₇	1	4	yes	—
412	C₂×C₄	C₂²×D₄×C₄₇	1	8	yes	—
413	C₂×C₄	C₂²×D₄×C₄₇	1	4	yes	—
414	C₂×C₄	C₂×#41×C₄₇	1	4	yes	—
415	C₂×C₄	C₂×#41×C₄₇	1	4	yes	—
416	C₂×C₄	C₂×#41×C₄₇	1	4	yes	—
417	C₂²	—	2	8	no	—
418	C₂²	—	2	8	no	—
419	C₂²	—	2	8	no	—
420	C₂²	—	4	32	no	—
421	C₂²	—	4	4	no	—
422	C₂²	—	4	4	no	—
423	C₂²	—	4	4	no	—
424	C₂²	—	4	4	no	—
425	C₂²	C₂×#117×C₄₇	1	4	yes	—
426	C₂²	C₂×C₄×D₄×C₄₇	1	4	yes	—
427	C₂²	C₄₇×#520	1	4	yes	—
428	C₂²	C₄₇×#547	1	4	yes	—
429	C₂²	C₄₇×#550	1	4	yes	—
430	C₂²	C₄₇×#555	1	4	yes	—
431	C₂²	C₄₇×#556	1	4	yes	—
432	C₂²	C₂⁴×C₄×C₄₇	1	1	yes	characteristic
433	C₂²	C₂³×D₄×C₄₇	1	1	yes	characteristic
434	C₂²	C₂²×#41×C₄₇	1	1	yes	characteristic
435	C₂²	C₂²×#41×C₄₇	1	1	yes	characteristic
436	C₂²	C₂²×#41×C₄₇	1	1	yes	characteristic
437	C₂²	C₂×H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
438	C₂²	C₂×2₋¹⁺⁴×C₄₇	1	1	yes	characteristic
439	C₄	—	2	8	no	—
440	C₄	—	2	4	no	—
441	C₄	—	2	8	no	—
442	C₄	—	4	8	no	—
443	C₄	—	4	4	no	—
444	C₄	C₂×#117×C₄₇	1	8	yes	complemented
445	C₄	C₂×C₄×D₄×C₄₇	1	8	yes	—
446	C₂	—	4	8	no	—
447	C₂	C₄₇×#2354	1	8	yes	direct product factor
448	C₂	C₂²×#117×C₄₇	1	1	yes	characteristic
449	C₂	C₂²×C₄×D₄×C₄₇	1	1	yes	characteristic
450	C₂	C₂×C₄₇×#520	1	1	yes	characteristic
451	C₂	C₂×C₄₇×#547	1	1	yes	characteristic
452	C₂	C₂×C₄₇×#550	1	1	yes	characteristic
453	C₂	C₂×C₄₇×#555	1	1	yes	characteristic
454	C₂	C₂×C₄₇×#556	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₄₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₄₇
Frattini, cosocle	C₂³	C₂⁵×C₄₇
prime radical	C₂³×D₄×C₄₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#117×C₄₇	C₂	semidirect
3. C₄₇×#2354	C₂	direct (N × H)
5. C₂×C₄₇×#388	C₂	semidirect
6. C₂×C₄₇×#384	C₂	semidirect
7. C₂×C₄₇×#380	C₂	semidirect
8. C₂×C₄×C₄⋊C₄×C₄₇	C₂	semidirect
9. C₂×C₄×C₄⋊C₄×C₄₇	C₂	semidirect
12. C₂×C₄×D₄×C₄₇	C₄	semidirect
15. C₂×#117×C₄₇	C₂²	semidirect
29. C₄₇×#388	C₂²	semidirect
30. C₄₇×#384	C₂²	semidirect
31. C₄₇×#380	C₂²	semidirect
32. C₄×C₄⋊C₄×C₄₇	C₂²	semidirect
33. C₄×C₄⋊C₄×C₄₇	C₂²	semidirect
52. C₄×D₄×C₄₇	C₂×C₄	semidirect
59. C₂×C₄⋊C₄×C₄₇	D₄	semidirect
130. C₄⋊C₄×C₄₇	C₂×D₄	semidirect
178. C₂×C₄×C₄₇	#117	semidirect
189. C₂×#2354	C₄₇	direct (N × H)
217. C₄×C₄₇	C₂×#117	semidirect
220. C₂²×#117	C₂×C₄₇	semidirect
221. #2354	C₂×C₄₇	direct (N × H)
223. C₂×#388	C₂×C₄₇	semidirect
224. C₂×#384	C₂×C₄₇	semidirect
225. C₂×#380	C₂×C₄₇	semidirect
226. C₂×C₄×C₄⋊C₄	C₂×C₄₇	semidirect
227. C₂×C₄×C₄⋊C₄	C₂×C₄₇	semidirect
229. C₂×C₄₇	#2354	direct (N × H)
239. C₂×C₄×D₄	C₄×C₄₇	semidirect
242. C₂×#117	C₂²×C₄₇	semidirect
256. #388	C₂²×C₄₇	semidirect
257. #384	C₂²×C₄₇	semidirect
258. #380	C₂²×C₄₇	semidirect
259. C₄×C₄⋊C₄	C₂²×C₄₇	semidirect
260. C₄×C₄⋊C₄	C₂²×C₄₇	semidirect
263. C₄₇	C₂×#2354	direct (N × H)
280. C₄×D₄	C₂×C₄×C₄₇	semidirect
287. C₂×C₄⋊C₄	D₄×C₄₇	semidirect
358. C₄⋊C₄	C₂×D₄×C₄₇	semidirect
406. C₂×C₄	#117×C₄₇	semidirect
444. C₄	C₂×#117×C₄₇	semidirect
447. C₂	C₄₇×#2354	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₄₇	2
C₂²×#117×C₄₇	2
C₄₇×#2354	2
C₂×C₄₇×#390	2
C₂×C₄₇×#388	2
C₂×C₄₇×#384	2
C₂×C₄₇×#380	2
C₂×C₄×C₄⋊C₄×C₄₇	2
C₂×#2354	47

Subgroups — C2×C47×#2354 #26073815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#2354 #26073815