Group #31897133 — subgroups

Subgroups — #31897133

Order 18176.

402 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986402	C₂	1	1	yes	maximal, characteristic
2	#22589330	C₂	1	1	yes	maximal, characteristic
3	#22589328	C₂	1	1	yes	maximal, characteristic
4	#22587095	C₂	1	1	yes	maximal, characteristic
5	#22587024	C₂	1	1	yes	maximal, characteristic
6	#22586630	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1549	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556128	C₂²	1	1	yes	characteristic
9	#17986402	—	2	1	no	—
10	#17986402	—	2	1	no	—
11	#17986402	—	2	2	no	—
12	C₂×#14556061	C₄	1	4	yes	—
13	#17986023	C₂²	1	1	yes	characteristic
14	C₄×#11765849	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
16	#17985768	—	2	2	no	—
17	#17985767	—	2	2	no	—
18	#17985749	—	2	2	no	—
19	#17985746	—	2	2	no	—
20	#17985721	—	2	1	no	—
21	#17985720	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	#431×C₇₁	C₂²	1	1	yes	characteristic
24	#423×C₇₁	C₂²	1	1	yes	characteristic
25	#332×C₇₁	—	2	2	no	—
26	C₂²×Dic₁₄₂	—	2	1	no	—
27	C₂×#11765849	—	2	1	no	—
28	C₂×#11765849	D₄	1	2	yes	—
29	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
30	#14556128	—	2	1	no	—
31	#14556128	—	2	1	no	—
32	#14556128	—	2	1	no	—
33	#14556128	—	2	1	no	—
34	C₂×C₄⋊Dic₇₁	—	2	1	no	—
35	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
36	#14556061	—	2	4	no	—
37	#14556061	—	2	4	no	—
38	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
39	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
40	#14556039	—	2	1	no	—
41	C₄×C₇₁⋊C₈	—	2	1	no	—
42	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
43	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
44	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
45	C₂²×Q₈×C₇₁	—	2	1	no	—
46	C₂×M₁₆×C₇₁	—	2	1	no	—
47	C₄⋊Q₈×C₇₁	—	2	1	no	—
48	C₄⋊Q₈×C₇₁	—	2	1	no	—
49	C₄⋊Q₈×C₇₁	—	2	1	no	—
50	C₄⋊Q₈×C₇₁	—	2	1	no	—
51	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₇₁	—	2	1	no	—
55	C₄⋊C₈×C₇₁	—	2	1	no	—
56	C₄⋊C₈×C₇₁	D₄	1	2	yes	—
57	C₂²⋊C₈×C₇₁	—	2	1	no	—
58	C₄×C₈×C₇₁	—	2	1	no	—
59	C₂×Dic₁₄₂	—	4	1	no	—
60	C₂×Dic₁₄₂	—	4	1	no	—
61	C₂×Dic₁₄₂	—	8	2	no	—
62	C₂²×Dic₇₁	—	2	2	no	—
63	#11765849	—	2	2	no	—
64	#11765849	—	2	2	no	—
65	#11765849	—	4	2	no	—
66	C₂×C₇₁⋊C₈	—	2	4	no	—
67	C₂×C₇₁⋊C₈	—	2	2	no	—
68	C₂×C₇₁⋊C₈	—	4	1	no	—
69	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
71	C₄⋊Dic₇₁	—	2	2	no	—
72	C₄⋊Dic₇₁	—	2	2	no	—
73	C₄⋊Dic₇₁	—	4	1	no	—
74	C₄⋊Dic₇₁	—	4	1	no	—
75	C₂×Q₈×C₇₁	—	4	1	no	—
76	C₂×Q₈×C₇₁	—	4	1	no	—
77	C₂×Q₈×C₇₁	—	8	2	no	—
78	C₂²×C₄×C₇₁	—	2	2	no	—
79	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
82	M₁₆×C₇₁	—	4	2	no	—
83	C₂×C₈×C₇₁	—	2	2	no	—
84	C₂×C₈×C₇₁	—	4	1	no	—
85	C₄⋊C₄×C₇₁	—	2	2	no	—
86	C₄⋊C₄×C₇₁	—	2	2	no	—
87	C₄⋊C₄×C₇₁	—	4	1	no	—
88	C₄⋊C₄×C₇₁	—	4	1	no	—
89	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₇₁	#41	1	1	yes	characteristic
93	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
94	Q₈×C₇₁	—	8	2	no	—
95	Q₈×C₇₁	—	8	2	no	—
96	C₂×C₄×C₇₁	—	2	2	no	—
97	C₂×C₄×C₇₁	—	2	2	no	—
98	C₂×C₄×C₇₁	—	4	2	no	—
99	C₂×C₄×C₇₁	—	8	2	no	—
100	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
103	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
104	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
105	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
106	C₂×C₄×C₇₁	#137	1	2	yes	—
107	C₂×C₄×C₇₁	#137	1	2	yes	—
108	C₂×Dic₇₁	—	4	2	no	—
109	C₂×Dic₇₁	—	8	2	no	—
110	Dic₁₄₂	—	8	2	no	—
111	Dic₁₄₂	—	8	2	no	—
112	C₈×C₇₁	—	4	2	no	—
113	C₈×C₇₁	—	8	1	no	—
114	C₇₁⋊C₈	—	2	2	no	—
115	C₇₁⋊C₈	—	4	2	no	—
116	C₇₁⋊C₈	—	8	1	no	—
117	C₇₁⋊C₈	#101	1	4	yes	—
118	C₂²×C₇₁	—	2	1	no	—
119	C₂²×C₇₁	—	2	1	no	—
120	C₂²×C₇₁	#390	1	1	yes	characteristic
121	C₂²×C₇₁	#441	1	2	yes	—
122	C₄×C₇₁	—	2	1	no	—
123	C₄×C₇₁	—	2	1	no	—
124	C₄×C₇₁	—	8	2	no	—
125	C₄×C₇₁	C₂×#101	1	2	yes	—
126	C₄×C₇₁	#470	1	2	yes	—
127	C₄×C₇₁	#472	1	2	yes	—
128	C₄×C₇₁	#497	1	1	yes	characteristic
129	C₄×C₇₁	#497	1	1	yes	characteristic
130	Dic₇₁	—	8	2	no	—
131	#9810	—	71	1	no	maximal, Sylow
132	C₂×C₇₁	—	2	2	no	—
133	C₂×C₇₁	#1966	1	1	yes	characteristic
134	C₂×C₇₁	#1971	1	1	yes	characteristic
135	C₂×C₇₁	#1978	1	1	yes	characteristic
136	C₂×#500	—	71	1	no	—
137	#1997	—	71	1	no	—
138	#1997	—	71	1	no	—
139	#1566	—	71	1	no	—
140	#1549	—	71	1	no	—
141	#1549	D₇₁	1	1	yes	characteristic
142	#1451	—	71	1	no	—
143	C₇₁	#9810	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄⋊Q₈	—	71	1	no	—
145	C₂×C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
146	#500	—	142	1	no	—
147	#500	—	142	1	no	—
148	#500	—	142	2	no	—
149	#431	C₂×D₇₁	1	1	yes	characteristic
150	C₂×#107	—	71	4	no	—
151	#423	—	71	1	no	—
152	#423	C₂×D₇₁	1	1	yes	characteristic
153	C₄×M₁₆	—	71	1	no	—
154	C₂×C₄×C₈	—	71	1	no	—
155	#343	—	142	2	no	—
156	#343	—	142	2	no	—
157	#334	—	142	2	no	—
158	#332	—	2	2	no	—
159	#332	—	142	2	no	—
160	#322	—	142	1	no	—
161	#322	—	142	1	no	—
162	C₂²×Q₈	—	2	1	no	—
163	C₂²×Q₈	—	142	1	no	—
164	C₂×M₁₆	—	2	1	no	—
165	C₂×M₁₆	—	71	2	no	—
166	C₂×M₁₆	—	142	1	no	—
167	C₂²×C₈	—	71	2	no	—
168	C₄⋊Q₈	—	2	1	no	—
169	C₄⋊Q₈	—	2	1	no	—
170	C₄⋊Q₈	—	2	1	no	—
171	C₄⋊Q₈	—	2	1	no	—
172	C₄⋊Q₈	—	142	1	no	—
173	C₄⋊Q₈	—	142	1	no	—
174	C₄⋊Q₈	—	142	1	no	—
175	C₄⋊Q₈	—	142	1	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	71	2	no	—
178	C₂×C₄⋊C₄	—	142	1	no	—
179	C₂×C₄⋊C₄	C₄×D₇₁	1	2	yes	—
180	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
181	#107	—	142	4	no	—
182	#107	—	142	4	no	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	142	1	no	—
186	C₄⋊C₈	—	142	1	no	—
187	C₄⋊C₈	C₂²⋊D₇₁	1	2	yes	—
188	C₂²⋊C₈	—	2	1	no	—
189	#97	—	142	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	71	1	no	—
192	C₄×C₈	—	71	1	no	—
193	C₄×C₈	—	71	2	no	—
194	C₄×C₈	—	142	1	no	—
195	C₂×Q₈	—	4	1	no	—
196	C₂×Q₈	—	4	1	no	—
197	C₂×Q₈	—	8	2	no	—
198	C₂×Q₈	—	284	1	no	—
199	C₂×Q₈	—	284	1	no	—
200	C₂×Q₈	—	568	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	142	2	no	—
203	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
204	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
205	C₂²×C₄	#11765864	1	1	yes	characteristic
206	M₁₆	—	4	2	no	—
207	M₁₆	—	284	2	no	—
208	M₁₆	—	142	2	no	—
209	M₁₆	—	142	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	284	1	no	—
213	C₂×C₈	—	71	2	no	—
214	C₂×C₈	—	71	2	no	—
215	C₂×C₈	—	142	2	no	—
216	C₂×C₈	—	142	4	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	284	1	no	—
222	C₄⋊C₄	—	284	1	no	—
223	C₄⋊C₄	—	142	2	no	—
224	C₄⋊C₄	—	142	2	no	—
225	C₄²	D₄×D₇₁	1	1	yes	characteristic
226	C₄²	D₄×D₇₁	1	1	yes	characteristic
227	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
228	C₄²	#11765865	1	1	yes	characteristic
229	C₂³	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
230	Q₈	—	8	2	no	—
231	Q₈	—	8	2	no	—
232	Q₈	—	568	2	no	—
233	Q₈	—	568	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	284	2	no	—
239	C₂×C₄	—	568	2	no	—
240	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
241	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
242	C₂×C₄	#14556100	1	1	yes	characteristic
243	C₂×C₄	#14556101	1	1	yes	characteristic
244	C₂×C₄	#14556114	1	1	yes	characteristic
245	C₂×C₄	#14556124	1	1	yes	characteristic
246	C₂×C₄	#14556157	1	2	yes	—
247	C₂×C₄	#14556160	1	2	yes	—
248	C₈	—	4	2	no	—
249	C₈	—	8	1	no	—
250	C₈	—	284	2	no	—
251	C₈	—	568	1	no	—
252	C₈	—	71	4	no	—
253	C₈	—	142	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	#17985899	1	1	yes	characteristic
257	C₂²	#17986103	1	2	yes	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	8	2	no	—
261	C₄	—	568	2	no	—
262	C₄	#101×D₇₁	1	2	yes	—
263	C₄	#17986256	1	2	yes	—
264	C₄	#17986267	1	2	yes	—
265	C₄	#17986390	1	1	yes	characteristic
266	C₄	#17986392	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	#22589182	1	1	yes	characteristic
269	C₂	#22589208	1	1	yes	characteristic
270	C₂	#22589249	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986402

C₂

yes

maximal, characteristic

#22589330

C₂

yes

maximal, characteristic

#22589328

C₂

yes

maximal, characteristic

#22587095

C₂

yes

maximal, characteristic

#22587024

C₂

yes

maximal, characteristic

#22586630

C₂

yes

maximal, characteristic

C₇₁×#1549

C₂

yes

maximal, characteristic

C₂×#14556128

C₂²

yes

—

—

—

C₄

yes

—

#17986023

C₂²

yes

characteristic

C₄×#11765849

C₂²

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

—

—

—

—

—

C₂×C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

#431×C₇₁

C₂²

yes

characteristic

#423×C₇₁

C₂²

yes

characteristic

#332×C₇₁

—

C₂²×Dic₁₄₂

—

C₂×#11765849

—

C₂×#11765849

D₄

yes

—

C₂²×C₇₁⋊C₈

C₂×C₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

C₂×C₄

yes

—

#14556061

—

#14556061

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₄×C₇₁⋊C₈

D₄

yes

—

C₂²×Q₈×C₇₁

—

C₂×M₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

D₄

yes

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

#11765849

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂²⋊C₄

yes

—

C₂×C₇₁⋊C₈

C₂²⋊C₄

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

#41

yes

characteristic

C₂³×C₇₁

C₄×D₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

100

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

101

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

102

C₂×C₄×C₇₁

#121

yes

characteristic

103

C₂×C₄×C₇₁

#121

yes

characteristic

104

C₂×C₄×C₇₁

#124

yes

characteristic

105

C₂×C₄×C₇₁

#127

yes

characteristic

106

C₂×C₄×C₇₁

#137

yes

—

107

C₂×C₄×C₇₁

#137

yes

—

108

C₂×Dic₇₁

—

109

C₂×Dic₇₁

—

110

Dic₁₄₂

—

111

Dic₁₄₂

—

112

C₈×C₇₁

—

113

C₈×C₇₁

—

114

C₇₁⋊C₈

—

115

C₇₁⋊C₈

—

116

C₇₁⋊C₈

—

117

C₇₁⋊C₈

#101

yes

—

118

C₂²×C₇₁

—

119

C₂²×C₇₁

—

120

C₂²×C₇₁

#390

yes

characteristic

121

C₂²×C₇₁

#441

yes

—

122

C₄×C₇₁

—

123

C₄×C₇₁

—

124

C₄×C₇₁

—

125

C₄×C₇₁

C₂×#101

yes

—

126

C₄×C₇₁

#470

yes

—

127

C₄×C₇₁

#472

yes

—

128

C₄×C₇₁

#497

yes

characteristic

129

C₄×C₇₁

#497

yes

characteristic

130

Dic₇₁

—

131

#9810

—

maximal, Sylow

132

C₂×C₇₁

—

133

C₂×C₇₁

#1966

yes

characteristic

134

C₂×C₇₁

#1971

yes

characteristic

135

C₂×C₇₁

#1978

yes

characteristic

136

C₂×#500

—

137

#1997

—

138

#1997

—

139

#1566

—

140

#1549

—

141

#1549

D₇₁

yes

characteristic

142

#1451

—

143

C₇₁

#9810

yes

characteristic, complemented, Sylow

144

C₂×C₄⋊Q₈

—

145

C₂×C₄⋊Q₈

C₂×D₇₁

yes

characteristic

146

#500

—

142

—

147

#500

—

142

—

148

#500

—

142

—

149

#431

C₂×D₇₁

yes

characteristic

150

C₂×#107

—

151

#423

—

152

#423

C₂×D₇₁

yes

characteristic

153

C₄×M₁₆

—

154

C₂×C₄×C₈

—

155

#343

—

142

—

156

#343

—

142

—

157

#334

—

142

—

158

#332

—

159

#332

—

142

—

160

#322

—

142

—

161

#322

—

142

—

162

C₂²×Q₈

—

163

C₂²×Q₈

—

142

—

164

C₂×M₁₆

—

165

C₂×M₁₆

—

166

C₂×M₁₆

—

142

—

167

C₂²×C₈

—

168

C₄⋊Q₈

—

169

C₄⋊Q₈

—

170

C₄⋊Q₈

—

171

C₄⋊Q₈

—

172

C₄⋊Q₈

—

142

—

173

C₄⋊Q₈

—

142

—

174

C₄⋊Q₈

—

142

—

175

C₄⋊Q₈

—

142

—

176

C₂×C₄⋊C₄

—

177

C₂×C₄⋊C₄

—

178

C₂×C₄⋊C₄

—

142

—

179

C₂×C₄⋊C₄

C₄×D₇₁

yes

—

180

C₂×C₄²

C₂²×D₇₁

yes

characteristic

181

#107

—

142

—

182

#107

—

142

—

183

C₄⋊C₈

—

184

C₄⋊C₈

—

185

C₄⋊C₈

—

142

—

186

C₄⋊C₈

—

142

—

187

C₄⋊C₈

C₂²⋊D₇₁

yes

—

188

C₂²⋊C₈

—

189

#97

—

142

—

190

C₄×C₈

—

191

C₄×C₈

—

192

C₄×C₈

—

193

C₄×C₈

—

194

C₄×C₈

—

142

—

195

C₂×Q₈

—

196

C₂×Q₈

—

197

C₂×Q₈

—

198

C₂×Q₈

—

284

—

199

C₂×Q₈

—

284

—

200

C₂×Q₈

—

568

—

201

C₂²×C₄

—

202

C₂²×C₄

—

142

—

203

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

204

C₂²×C₄

D₄×D₇₁

yes

characteristic

205

C₂²×C₄

#11765864

yes

characteristic

206

M₁₆

—

207

M₁₆

—

284

—

208

M₁₆

—

142

—

209

M₁₆

—

142

—

210

C₂×C₈

—

211

C₂×C₈

—

212

C₂×C₈

—

284

—

213

C₂×C₈

—

214

C₂×C₈

—

215

C₂×C₈

—

142

—

216

C₂×C₈

—

142

—

217

C₄⋊C₄

—

218

C₄⋊C₄

—

219

C₄⋊C₄

—

220

C₄⋊C₄

—

221

C₄⋊C₄

—

284

—

222

C₄⋊C₄

—

284

—

223

C₄⋊C₄

—

142

—

224

C₄⋊C₄

—

142

—

225

C₄²

D₄×D₇₁

yes

characteristic

226

C₄²

D₄×D₇₁

yes

characteristic

227

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

228

C₄²

#11765865

yes

characteristic

229

C₂³

(C₂²⋊D₇₁)⋊C₄

yes

characteristic

230

Q₈

—

231

Q₈

—

232

Q₈

—

568

—

233

Q₈

—

568

—

234

C₂×C₄

—

235

C₂×C₄

—

236

C₂×C₄

—

237

C₂×C₄

—

238

C₂×C₄

—

284

—

239

C₂×C₄

—

568

—

240

C₂×C₄

C₂²⋊C₄×D₇₁

yes

characteristic

241

C₂×C₄

C₄×C₂²⋊D₇₁

yes

characteristic

242

C₂×C₄

#14556100

yes

characteristic

243

C₂×C₄

#14556101

yes

characteristic

244

C₂×C₄

#14556114

yes

characteristic

245

C₂×C₄

#14556124

yes

characteristic

246

C₂×C₄

#14556157

yes

—

247

C₂×C₄

#14556160

yes

—

248

C₈

—

249

C₈

—

250

C₈

—

284

—

251

C₈

—

568

—

252

C₈

—

253

C₈

—

142

—

254

C₂²

—

255

C₂²

—

256

C₂²

#17985899

yes

characteristic

257

C₂²

#17986103

yes

—

258

C₄

—

259

C₄

—

260

C₄

—

261

C₄

—

568

—

262

C₄

#101×D₇₁

yes

—

263

C₄

#17986256

yes

—

264

C₄

#17986267

yes

—

265

C₄

#17986390

yes

characteristic

266

C₄

#17986392

yes

characteristic

267

C₂

—

268

C₂

#22589182

yes

characteristic

269

C₂

#22589208

yes

characteristic

270

C₂

#22589249

yes

characteristic

	subgroup	G/H
center	C₂²	#17985899
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1549	C₂
socle	C₂²×C₇₁	#390
prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1549	D₇₁
hypocenter	C₇₁	#9810
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1549

subgroup

G/H

center

C₂²

#17985899

commutator

C₂²×C₄×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1549

C₂

socle

C₂²×C₇₁

#390

prime radical

C₂³×C₇₁

C₄×D₄

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1549

N	H ≅ G/N	type
143. C₇₁	#9810	semidirect

H ≅ G/N

type

143. C₇₁

#9810

semidirect

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

subgroup	index
C₂×#17986402	2
#22589330	2
#22589328	2
#22587095	2
#22587024	2
#22586630	2
C₇₁×#1549	2
#9810	71

subgroup

index

C₇₁×#1549

Subgroups — #31897133

Conjugacy classes of subgroups

Complements

Maximal subgroups