Group #31902091 — subgroups

Subgroups — #31902091

Order 18176.

306 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 97 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	#22589330	C₂	1	1	yes	maximal, characteristic
2	#22589113	C₂	1	1	yes	maximal, characteristic
3	#22587115	C₂	1	1	yes	maximal, characteristic
4	#22587093	C₂	1	1	yes	maximal, characteristic
5	C₂×#17985748	C₂	1	1	yes	maximal, characteristic
6	#22586214	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1548	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556063	—	2	1	no	—
9	#17986023	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄×#11765849	C₂²	1	1	yes	characteristic
13	#17985768	—	2	1	no	—
14	#17985768	—	2	1	no	—
15	#17985763	—	2	1	no	—
16	#17985752	—	2	2	no	—
17	#17985751	—	2	1	no	—
18	#17985749	—	2	1	no	—
19	#17985748	—	2	2	no	—
20	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
21	#431×C₇₁	C₂²	1	1	yes	characteristic
22	#423×C₇₁	C₂²	1	1	yes	characteristic
23	#333×C₇₁	—	2	1	no	—
24	#333×C₇₁	—	2	1	no	—
25	C₂×#11765849	—	2	1	no	—
26	C₂×#11765849	D₄	1	1	yes	characteristic
27	C₂×#11765849	D₄	1	1	yes	characteristic
28	C₂²×C₇₁⋊C₈	—	2	1	no	—
29	C₂²×C₇₁⋊C₈	—	2	1	no	—
30	#14556063	—	4	2	no	—
31	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
32	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
33	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
34	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
35	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
36	C₄⋊(C₇₁⋊C₈)	D₄	1	2	yes	—
37	#14556039	—	2	1	no	—
38	C₄×C₇₁⋊C₈	—	2	1	no	—
39	C₂²×Q₈×C₇₁	Q₈	1	2	yes	—
40	C₂×M₁₆×C₇₁	—	2	1	no	—
41	C₄⋊Q₈×C₇₁	—	2	1	no	—
42	C₄⋊Q₈×C₇₁	—	2	1	no	—
43	C₄⋊Q₈×C₇₁	—	2	1	no	—
44	C₄⋊Q₈×C₇₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
46	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
47	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₇₁	—	2	1	no	—
49	C₄⋊C₈×C₇₁	—	2	1	no	—
50	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
51	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
52	C₂²⋊C₈×C₇₁	—	2	1	no	—
53	C₄×C₈×C₇₁	—	2	1	no	—
54	#11765849	—	2	1	no	—
55	#11765849	—	2	1	no	—
56	#11765849	—	2	1	no	—
57	#11765849	—	4	1	no	—
58	#11765849	—	4	1	no	—
59	#11765849	Q₁₆	1	2	yes	—
60	C₂×C₇₁⋊C₈	—	2	1	no	—
61	C₂×C₇₁⋊C₈	—	2	1	no	—
62	C₂×C₇₁⋊C₈	—	2	2	no	—
63	C₂×C₇₁⋊C₈	—	4	1	no	—
64	C₂×C₇₁⋊C₈	—	4	1	no	—
65	C₂×C₇₁⋊C₈	—	4	1	no	—
66	C₂×C₇₁⋊C₈	—	4	1	no	—
67	C₂×C₇₁⋊C₈	—	8	1	no	—
68	C₂×Q₈×C₇₁	—	2	2	no	—
69	C₂×Q₈×C₇₁	—	2	2	no	—
70	C₂×Q₈×C₇₁	—	8	2	no	—
71	C₂²×C₄×C₇₁	—	2	2	no	—
72	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
75	M₁₆×C₇₁	—	4	1	no	—
76	M₁₆×C₇₁	—	4	1	no	—
77	C₂×C₈×C₇₁	—	2	1	no	—
78	C₂×C₈×C₇₁	—	2	1	no	—
79	C₂×C₈×C₇₁	—	4	1	no	—
80	C₄⋊C₄×C₇₁	—	4	1	no	—
81	C₄⋊C₄×C₇₁	—	4	1	no	—
82	C₄⋊C₄×C₇₁	—	4	1	no	—
83	C₄⋊C₄×C₇₁	—	4	1	no	—
84	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₇₁	#41	1	1	yes	characteristic
87	C₄²×C₇₁	#41	1	1	yes	characteristic
88	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
89	Q₈×C₇₁	—	8	2	no	—
90	Q₈×C₇₁	—	8	2	no	—
91	C₂×C₄×C₇₁	—	2	1	no	—
92	C₂×C₄×C₇₁	—	2	1	no	—
93	C₂×C₄×C₇₁	—	4	1	no	—
94	C₂×C₄×C₇₁	—	4	2	no	—
95	C₂×C₄×C₇₁	—	8	2	no	—
96	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
97	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
98	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
99	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₇₁	#126	1	1	yes	characteristic
102	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
104	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
106	C₈×C₇₁	—	4	1	no	—
107	C₈×C₇₁	—	4	1	no	—
108	C₈×C₇₁	—	8	1	no	—
109	C₇₁⋊C₈	—	2	2	no	—
110	C₇₁⋊C₈	—	4	2	no	—
111	C₇₁⋊C₈	—	4	1	no	—
112	C₇₁⋊C₈	—	8	1	no	—
113	C₇₁⋊C₈	—	8	1	no	—
114	C₂²×C₇₁	—	2	1	no	—
115	C₂²×C₇₁	—	2	1	no	—
116	C₂²×C₇₁	#396	1	1	yes	characteristic
117	C₂²×C₇₁	#477	1	1	yes	characteristic
118	C₂²×C₇₁	#479	1	1	yes	characteristic
119	C₄×C₇₁	—	2	1	no	—
120	C₄×C₇₁	—	2	1	no	—
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	—	8	2	no	—
123	C₄×C₇₁	#462	1	1	yes	characteristic
124	C₄×C₇₁	#463	1	1	yes	characteristic
125	C₄×C₇₁	#467	1	1	yes	characteristic
126	C₄×C₇₁	#468	1	1	yes	characteristic
127	C₄×C₇₁	#471	1	1	yes	characteristic
128	C₄×C₇₁	#473	1	1	yes	characteristic
129	#10605	—	71	1	no	maximal, Sylow
130	C₂×C₇₁	—	2	1	no	—
131	C₂×C₇₁	—	2	1	no	—
132	C₂×C₇₁	#2064	1	1	yes	characteristic
133	C₂×C₇₁	#2068	1	1	yes	characteristic
134	C₂×C₇₁	#2069	1	1	yes	characteristic
135	#1997	—	71	1	no	—
136	#1950	—	71	1	no	—
137	#1571	—	71	1	no	—
138	#1565	—	71	1	no	—
139	#1548	D₇₁	1	1	yes	characteristic
140	C₂×#333	—	71	1	no	—
141	#1292	—	71	1	no	—
142	C₇₁	#10605	1	1	yes	characteristic, complemented, Sylow
143	C₂×C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
144	#431	C₂×D₇₁	1	1	yes	characteristic
145	C₂×#108	—	142	1	no	—
146	#423	—	71	1	no	—
147	#423	C₂×D₇₁	1	1	yes	characteristic
148	C₂×C₄⋊C₈	—	71	1	no	—
149	C₂×C₄⋊C₈	—	71	1	no	—
150	C₄×M₁₆	—	71	1	no	—
151	#343	—	142	1	no	—
152	#343	—	142	1	no	—
153	#341	—	142	1	no	—
154	#335	—	142	2	no	—
155	#335	—	142	1	no	—
156	#334	—	142	1	no	—
157	#333	—	2	1	no	—
158	#333	—	2	1	no	—
159	#333	—	142	2	no	—
160	C₂²×Q₈	Dic₁₄₂	1	2	yes	—
161	C₂×M₁₆	—	2	1	no	—
162	C₂×M₁₆	—	71	1	no	—
163	C₂×M₁₆	—	71	1	no	—
164	C₂×M₁₆	—	142	1	no	—
165	C₂²×C₈	—	142	1	no	—
166	C₂²×C₈	—	142	1	no	—
167	C₄⋊Q₈	—	2	1	no	—
168	C₄⋊Q₈	—	2	1	no	—
169	C₄⋊Q₈	—	2	1	no	—
170	C₄⋊Q₈	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
174	#108	—	284	2	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	71	2	no	—
178	C₄⋊C₈	—	142	1	no	—
179	C₄⋊C₈	—	142	1	no	—
180	C₄⋊C₈	—	142	1	no	—
181	C₄⋊C₈	—	142	1	no	—
182	C₄⋊C₈	—	142	1	no	—
183	C₄⋊C₈	C₂²⋊D₇₁	1	1	yes	characteristic
184	C₄⋊C₈	C₂²⋊D₇₁	1	1	yes	characteristic
185	C₂²⋊C₈	—	2	1	no	—
186	#97	—	142	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	142	1	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	8	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
194	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
195	C₂²×C₄	#11765864	1	1	yes	characteristic
196	M₁₆	—	4	1	no	—
197	M₁₆	—	4	1	no	—
198	M₁₆	—	284	1	no	—
199	M₁₆	—	284	1	no	—
200	M₁₆	—	71	2	no	—
201	M₁₆	—	142	1	no	—
202	M₁₆	—	142	1	no	—
203	M₁₆	—	142	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	284	1	no	—
208	C₂×C₈	—	284	1	no	—
209	C₂×C₈	—	284	1	no	—
210	C₂×C₈	—	284	1	no	—
211	C₂×C₈	—	568	1	no	—
212	C₂×C₈	—	142	1	no	—
213	C₂×C₈	—	142	2	no	—
214	C₂×C₈	—	142	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄²	D₄×D₇₁	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
221	C₄²	#11765864	1	1	yes	characteristic
222	C₄²	#11765865	1	1	yes	characteristic
223	C₂³	#14556105	1	1	yes	characteristic
224	Q₈	—	8	2	no	—
225	Q₈	—	8	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	#14556099	1	1	yes	characteristic
232	C₂×C₄	#14556100	1	1	yes	characteristic
233	C₂×C₄	#14556101	1	1	yes	characteristic
234	C₂×C₄	#14556105	1	1	yes	characteristic
235	C₂×C₄	#14556107	1	1	yes	characteristic
236	C₂×C₄	#14556121	1	1	yes	characteristic
237	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
238	C₂×C₄	#14556151	1	1	yes	characteristic
239	C₂×C₄	#14556155	1	1	yes	characteristic
240	C₂×C₄	#14556157	1	1	yes	characteristic
241	C₈	—	4	1	no	—
242	C₈	—	4	1	no	—
243	C₈	—	8	1	no	—
244	C₈	—	284	2	no	—
245	C₈	—	284	1	no	—
246	C₈	—	568	1	no	—
247	C₈	—	568	1	no	—
248	C₈	—	142	2	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	#17985922	1	1	yes	characteristic
252	C₂²	#17986292	1	1	yes	characteristic
253	C₂²	#17986308	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	8	2	no	—
258	C₄	#17986208	1	1	yes	characteristic
259	C₄	#17986217	1	1	yes	characteristic
260	C₄	#17986237	1	1	yes	characteristic
261	C₄	#17986242	1	1	yes	characteristic
262	C₄	#17986264	1	1	yes	characteristic
263	C₄	#17986272	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	#22589707	1	1	yes	characteristic
267	C₂	#22589733	1	1	yes	characteristic
268	C₂	#22589738	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22589330

C₂

yes

maximal, characteristic

#22589113

C₂

yes

maximal, characteristic

#22587115

C₂

yes

maximal, characteristic

#22587093

C₂

yes

maximal, characteristic

C₂×#17985748

C₂

yes

maximal, characteristic

#22586214

C₂

yes

maximal, characteristic

C₇₁×#1548

C₂

yes

maximal, characteristic

C₂×#14556063

—

#17986023

C₂²

yes

characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄×#11765849

C₂²

yes

—

—

—

—

—

—

—

C₂×C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

#431×C₇₁

C₂²

yes

characteristic

#423×C₇₁

C₂²

yes

characteristic

#333×C₇₁

—

#333×C₇₁

—

C₂×#11765849

—

C₂×#11765849

D₄

yes

characteristic

C₂×#11765849

D₄

yes

characteristic

C₂²×C₇₁⋊C₈

—

C₂²×C₇₁⋊C₈

—

#14556063

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

D₄

yes

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₂²×Q₈×C₇₁

Q₈

yes

—

C₂×M₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

D₄

yes

characteristic

C₄⋊C₈×C₇₁

D₄

yes

characteristic

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

—

—

—

—

—

Q₁₆

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×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

C₄²×C₇₁

yes

C₄²×C₇₁

yes

C₂³×C₇₁

C₂²⋊Q₈

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

100

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

101

C₂×C₄×C₇₁

#126

yes

characteristic

102

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

103

C₂×C₄×C₇₁

#135

yes

characteristic

104

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

characteristic

105

C₂×C₄×C₇₁

#137

yes

characteristic

106

C₈×C₇₁

—

107

C₈×C₇₁

—

108

C₈×C₇₁

—

109

C₇₁⋊C₈

—

110

C₇₁⋊C₈

—

111

C₇₁⋊C₈

—

112

C₇₁⋊C₈

—

113

C₇₁⋊C₈

—

114

C₂²×C₇₁

—

115

C₂²×C₇₁

—

116

C₂²×C₇₁

#396

yes

characteristic

117

C₂²×C₇₁

#477

yes

characteristic

118

C₂²×C₇₁

#479

yes

characteristic

119

C₄×C₇₁

—

120

C₄×C₇₁

—

121

C₄×C₇₁

—

122

C₄×C₇₁

—

123

C₄×C₇₁

#462

yes

characteristic

124

C₄×C₇₁

#463

yes

characteristic

125

C₄×C₇₁

#467

yes

characteristic

126

C₄×C₇₁

#468

yes

characteristic

127

C₄×C₇₁

#471

yes

characteristic

128

C₄×C₇₁

yes

129

—

130

C₂×C₇₁

—

131

C₂×C₇₁

—

132

C₂×C₇₁

#2064

yes

characteristic

133

C₂×C₇₁

#2068

yes

characteristic

134

C₂×C₇₁

#2069

yes

characteristic

135

#1997

—

136

#1950

—

137

#1571

—

138

#1565

—

139

#1548

D₇₁

yes

characteristic

140

C₂×#333

—

141

#1292

—

142

C₇₁

#10605

yes

characteristic, complemented, Sylow

143

C₂×C₄⋊Q₈

C₂×D₇₁

yes

characteristic

144

#431

C₂×D₇₁

yes

characteristic

145

C₂×#108

—

142

—

146

#423

—

147

#423

C₂×D₇₁

yes

characteristic

148

C₂×C₄⋊C₈

—

149

C₂×C₄⋊C₈

—

150

C₄×M₁₆

—

151

#343

—

142

—

152

#343

—

142

—

153

#341

—

142

—

154

#335

—

142

—

155

#335

—

142

—

156

#334

—

142

—

157

#333

—

158

#333

—

159

#333

—

142

—

160

C₂²×Q₈

Dic₁₄₂

yes

—

161

C₂×M₁₆

—

162

C₂×M₁₆

—

163

C₂×M₁₆

—

164

C₂×M₁₆

—

142

—

165

C₂²×C₈

—

142

—

166

C₂²×C₈

—

142

—

167

C₄⋊Q₈

—

168

C₄⋊Q₈

—

169

C₄⋊Q₈

—

170

C₄⋊Q₈

—

171

C₂×C₄⋊C₄

—

172

C₂×C₄⋊C₄

—

173

C₂×C₄²

C₂²×D₇₁

yes

characteristic

174

#108

—

284

—

175

C₄⋊C₈

—

176

C₄⋊C₈

—

177

C₄⋊C₈

—

178

C₄⋊C₈

—

142

—

179

C₄⋊C₈

—

142

—

180

C₄⋊C₈

—

142

—

181

C₄⋊C₈

—

142

—

182

C₄⋊C₈

—

142

—

183

C₄⋊C₈

C₂²⋊D₇₁

yes

characteristic

184

C₄⋊C₈

C₂²⋊D₇₁

yes

characteristic

185

C₂²⋊C₈

—

186

#97

—

142

—

187

C₄×C₈

—

188

C₄×C₈

—

142

—

189

C₂×Q₈

—

190

C₂×Q₈

—

191

C₂×Q₈

—

192

C₂²×C₄

—

193

C₂²×C₄

D₄×D₇₁

yes

characteristic

194

C₂²×C₄

C₂×Dic₁₄₂

yes

characteristic

195

C₂²×C₄

#11765864

yes

characteristic

196

M₁₆

—

197

M₁₆

—

198

M₁₆

—

284

—

199

M₁₆

—

284

—

200

M₁₆

—

201

M₁₆

—

142

—

202

M₁₆

—

142

—

203

M₁₆

—

142

—

204

C₂×C₈

—

205

C₂×C₈

—

206

C₂×C₈

—

207

C₂×C₈

—

284

—

208

C₂×C₈

—

284

—

209

C₂×C₈

—

284

—

210

C₂×C₈

—

284

—

211

C₂×C₈

—

568

—

212

C₂×C₈

—

142

—

213

C₂×C₈

—

142

—

214

C₂×C₈

—

142

—

215

C₄⋊C₄

—

216

C₄⋊C₄

—

217

C₄⋊C₄

—

218

C₄⋊C₄

—

219

C₄²

D₄×D₇₁

yes

characteristic

220

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

221

C₄²

#11765864

yes

characteristic

222

C₄²

#11765865

yes

characteristic

223

C₂³

#14556105

yes

characteristic

224

Q₈

—

225

Q₈

—

226

C₂×C₄

—

227

C₂×C₄

—

228

C₂×C₄

—

229

C₂×C₄

—

230

C₂×C₄

—

231

C₂×C₄

#14556099

yes

characteristic

232

C₂×C₄

#14556100

yes

characteristic

233

C₂×C₄

#14556101

yes

characteristic

234

C₂×C₄

#14556105

yes

characteristic

235

C₂×C₄

#14556107

yes

characteristic

236

C₂×C₄

#14556121

yes

characteristic

237

C₂×C₄

Q₁₆×D₇₁

yes

characteristic

238

C₂×C₄

#14556151

yes

characteristic

239

C₂×C₄

#14556155

yes

characteristic

240

C₂×C₄

#14556157

yes

characteristic

241

C₈

—

242

C₈

—

243

C₈

—

244

C₈

—

284

—

245

C₈

—

284

—

246

C₈

—

568

—

247

C₈

—

568

—

248

C₈

—

142

—

249

C₂²

—

250

C₂²

—

251

C₂²

#17985922

yes

characteristic

252

C₂²

#17986292

yes

characteristic

253

C₂²

#17986308

yes

characteristic

254

C₄

—

255

C₄

—

256

C₄

—

257

C₄

—

258

C₄

#17986208

yes

characteristic

259

C₄

#17986217

yes

characteristic

260

C₄

#17986237

yes

characteristic

261

C₄

#17986242

yes

characteristic

262

C₄

#17986264

yes

characteristic

263

C₄

#17986272

yes

characteristic

264

C₂

—

265

C₂

—

266

C₂

#22589707

yes

characteristic

267

C₂

#22589733

yes

characteristic

268

C₂

#22589738

yes

characteristic

	subgroup	G/H
center	C₂²	#17985922
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1548	C₂
socle	C₂²×C₇₁	#396
prime radical	C₂³×C₇₁	C₂²⋊Q₈
hypercenter	#1548	D₇₁
hypocenter	C₇₁	#10605
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1548

subgroup

G/H

center

C₂²

#17985922

commutator, cosocle

C₂×C₄²×C₇₁

C₂³

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1548

C₂

socle

C₂²×C₇₁

#396

prime radical

C₂³×C₇₁

C₂²⋊Q₈

D₇₁

C₇₁

G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1548

N	H ≅ G/N	type
142. C₇₁	#10605	semidirect

H ≅ G/N

type

142. C₇₁

#10605

semidirect

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

subgroup	index
#22589330	2
#22589113	2
#22587115	2
#22587093	2
C₂×#17985748	2
#22586214	2
C₇₁×#1548	2
#10605	71

subgroup

index

C₇₁×#1548

Subgroups — #31902091

Conjugacy classes of subgroups

Complements

Maximal subgroups