Group #35319682 — subgroups

Subgroups — #35319682

Order 21248.

422 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 127 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×#19127807	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127518	C₂	1	1	yes	maximal, characteristic
3	#24909182	C₂	1	1	yes	maximal, characteristic
4	C₂×#19127478	C₂	1	1	yes	maximal, characteristic
5	#24907349	C₂	1	1	yes	maximal, characteristic
6	#24905561	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#1386	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₃₂	C₂²	1	1	yes	characteristic
9	C₂×#14581757	—	2	1	no	—
10	C₂×#14581732	C₂²	1	1	yes	characteristic
11	#19127807	—	2	2	no	—
12	#19127807	C₄	1	4	yes	complemented
13	C₂×#14581728	C₂²	1	1	yes	characteristic
14	C₂×#14581715	—	2	1	no	—
15	#19127657	—	2	1	no	—
16	C₂×#14581702	C₂²	1	1	yes	characteristic
17	#19127518	C₄	1	4	yes	complemented
18	#19127504	C₄	1	4	yes	complemented
19	#19127478	—	2	2	no	—
20	#19127460	—	2	1	no	—
21	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
22	#427×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×#106×C₈₃	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₆₆	—	2	1	no	—
25	C₂×Dic₃₃₂	—	2	1	no	—
26	C₂×Dic₃₃₂	—	2	1	no	—
27	C₂×Dic₃₃₂	—	2	1	no	—
28	C₂×Dic₃₃₂	C₂×C₄	1	2	yes	—
29	C₂×Dic₃₃₂	C₂×C₄	1	2	yes	—
30	C₂×Dic₃₃₂	C₂×C₄	1	2	yes	—
31	C₂×#13050574	—	2	1	no	—
32	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
33	#14581757	—	2	4	no	—
34	#14581757	—	4	2	no	—
35	C₂×Dic₈₃⋊C₄	—	4	1	no	—
36	C₂×C₄×Dic₈₃	—	4	1	no	—
37	C₂×C₄×Dic₈₃	D₄	1	1	yes	characteristic
38	#14581732	—	2	1	no	—
39	#14581732	—	2	1	no	—
40	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
41	#14581728	—	2	1	no	—
42	#14581728	C₂×C₄	1	2	yes	—
43	#14581715	—	2	4	no	—
44	#14581703	—	4	1	no	—
45	#14581702	C₂×C₄	1	2	yes	—
46	#14581702	D₄	1	2	yes	—
47	C₄×C₈₃⋊C₈	—	4	1	no	—
48	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
49	#117×C₈₃	—	2	1	no	—
50	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
51	M₃₂×C₈₃	—	2	1	no	—
52	M₃₂×C₈₃	C₂×C₄	1	2	yes	—
53	C₂×C₁₆×C₈₃	—	2	1	no	—
54	#107×C₈₃	C₂×C₄	1	2	yes	—
55	#106×C₈₃	—	2	1	no	—
56	#106×C₈₃	—	2	1	no	—
57	#106×C₈₃	C₂×C₄	1	2	yes	—
58	C₂×Dic₁₆₆	—	2	2	no	—
59	C₂×Dic₁₆₆	—	2	2	no	—
60	C₂×Dic₁₆₆	—	2	2	no	—
61	C₂×Dic₁₆₆	—	4	1	no	—
62	C₂×Dic₁₆₆	—	4	1	no	—
63	C₂×Dic₁₆₆	—	4	1	no	—
64	C₂²×Dic₈₃	—	2	1	no	—
65	C₂²×Dic₈₃	—	4	1	no	—
66	Dic₃₃₂	—	2	2	no	—
67	Dic₃₃₂	—	2	2	no	—
68	Dic₃₃₂	—	2	2	no	—
69	Dic₃₃₂	C₄²	1	4	yes	—
70	#13050574	—	2	2	no	—
71	#13050574	—	4	1	no	—
72	C₂×C₈₃⋊C₈	—	2	1	no	—
73	C₂×C₈₃⋊C₈	—	2	1	no	—
74	C₂×C₈₃⋊C₈	—	4	1	no	—
75	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	2	yes	—
76	Dic₈₃⋊C₄	—	4	4	no	—
77	C₄×Dic₈₃	—	2	1	no	—
78	C₄×Dic₈₃	—	4	4	no	—
79	C₄×Dic₈₃	C₂²⋊C₄	1	2	yes	—
80	C₈₃⋊C₁₆	—	2	2	no	—
81	C₂²×C₄×C₈₃	—	4	1	no	—
82	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
89	C₄⋊C₄×C₈₃	—	2	2	no	—
90	C₄⋊C₄×C₈₃	—	2	2	no	—
91	C₄⋊C₄×C₈₃	—	4	1	no	—
92	C₂²⋊C₄×C₈₃	—	4	1	no	—
93	C₄²×C₈₃	—	4	1	no	—
94	C₁₆×C₈₃	—	2	2	no	—
95	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₈₃	—	4	2	no	—
97	C₂×C₄×C₈₃	—	4	2	no	—
98	C₂×C₄×C₈₃	—	8	2	no	—
99	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
105	C₂×Dic₈₃	—	2	2	no	—
106	C₂×Dic₈₃	—	4	2	no	—
107	C₂×Dic₈₃	—	4	2	no	—
108	C₂×Dic₈₃	—	4	2	no	—
109	C₂×Dic₈₃	—	4	2	no	—
110	Dic₁₆₆	—	2	4	no	—
111	Dic₁₆₆	—	4	2	no	—
112	Dic₁₆₆	—	4	2	no	—
113	Dic₁₆₆	—	4	2	no	—
114	C₈×C₈₃	C₂×C₄²	1	1	yes	characteristic
115	C₈×C₈₃	#117	1	1	yes	characteristic
116	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
117	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
118	C₈₃⋊C₈	—	2	2	no	—
119	C₈₃⋊C₈	—	4	2	no	—
120	C₂²×C₈₃	—	2	2	no	—
121	C₂²×C₈₃	#418	1	1	yes	characteristic
122	C₂²×C₈₃	#440	1	1	yes	characteristic
123	C₂²×C₈₃	#442	1	1	yes	characteristic
124	C₄×C₈₃	—	8	2	no	—
125	C₄×C₈₃	—	8	2	no	—
126	C₄×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₈₃	#417	1	1	yes	characteristic
128	C₄×C₈₃	#440	1	1	yes	characteristic
129	C₄×C₈₃	#442	1	1	yes	characteristic
130	Dic₈₃	—	4	2	no	—
131	Dic₈₃	—	4	4	no	—
132	#12648	—	83	1	no	maximal, Sylow
133	C₂×C₈₃	—	2	1	no	—
134	C₂×C₈₃	—	2	1	no	—
135	C₂×C₈₃	#1762	1	1	yes	characteristic
136	C₂×C₈₃	#2180	1	2	yes	—
137	C₂×#441	—	83	1	no	—
138	C₂×#362	—	83	1	no	—
139	#2141	—	83	1	no	—
140	C₂×#347	—	83	1	no	—
141	#1773	—	83	1	no	—
142	#1386	D₈₃	1	1	yes	characteristic
143	#1383	—	83	1	no	—
144	C₈₃	#12648	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₁₆	—	83	1	no	—
146	C₂×C₄×Q₈	—	166	1	no	—
147	C₂×M₃₂	—	83	1	no	—
148	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
149	#441	—	83	4	no	—
150	#441	—	166	2	no	—
151	C₂×#108	—	83	1	no	—
152	#427	C₂×D₈₃	1	1	yes	characteristic
153	C₂×#106	C₂×D₈₃	1	1	yes	characteristic
154	C₂×#103	—	166	1	no	—
155	#405	—	166	1	no	—
156	C₂×#97	—	83	1	no	—
157	#362	—	83	4	no	—
158	#358	—	83	4	no	—
159	#347	—	166	2	no	—
160	#339	—	166	1	no	—
161	C₂²×Q₈	—	166	1	no	—
162	C₂×Q₁₆	—	83	2	no	—
163	C₂×Q₁₆	—	83	2	no	—
164	C₂×Q₁₆	—	83	2	no	—
165	C₂×Q₁₆	—	166	1	no	—
166	C₂×Q₁₆	—	166	1	no	—
167	C₂×Q₁₆	—	166	1	no	—
168	C₂×M₁₆	—	166	1	no	—
169	C₂²×C₈	—	83	1	no	—
170	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
171	C₄×Q₈	—	332	2	no	—
172	C₄×Q₈	—	166	4	no	—
173	#117	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	332	1	no	—
176	C₂×C₄²	—	332	1	no	—
177	C₂×C₄²	—	83	1	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	—	166	1	no	—
180	M₃₂	—	166	1	no	—
181	M₃₂	C₄×D₈₃	1	2	yes	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	83	2	no	—
184	#108	—	83	2	no	—
185	#108	—	166	1	no	—
186	#107	C₄×D₈₃	1	2	yes	—
187	#106	—	2	1	no	—
188	#106	—	2	1	no	—
189	#106	C₄×D₈₃	1	2	yes	—
190	#103	—	166	4	no	—
191	#97	—	332	1	no	—
192	#97	—	83	2	no	—
193	#97	—	83	2	no	—
194	C₄×C₈	—	332	1	no	—
195	C₂×Q₈	—	332	1	no	—
196	C₂×Q₈	—	332	1	no	—
197	C₂×Q₈	—	332	1	no	—
198	C₂×Q₈	—	166	2	no	—
199	C₂×Q₈	—	166	2	no	—
200	C₂×Q₈	—	166	2	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	332	1	no	—
203	C₂²×C₄	—	166	1	no	—
204	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
205	Q₁₆	—	83	4	no	—
206	Q₁₆	—	166	2	no	—
207	Q₁₆	—	166	2	no	—
208	Q₁₆	—	166	2	no	—
209	M₁₆	—	332	1	no	—
210	M₁₆	—	166	2	no	—
211	C₂×C₈	—	332	1	no	—
212	C₂×C₈	—	83	2	no	—
213	C₂×C₈	—	166	1	no	—
214	C₂×C₈	—	166	1	no	—
215	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
217	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
218	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
219	C₂×C₈	#13050589	1	1	yes	characteristic
220	C₂×C₈	#13050591	1	1	yes	characteristic
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	332	4	no	—
225	C₂²⋊C₄	—	4	1	no	—
226	C₄²	—	4	1	no	—
227	C₄²	—	332	4	no	—
228	C₄²	—	83	2	no	—
229	C₄²	—	166	1	no	—
230	C₁₆	—	2	2	no	—
231	C₁₆	—	166	2	no	—
232	C₂³	#14581816	1	1	yes	characteristic
233	Q₈	—	332	2	no	—
234	Q₈	—	332	2	no	—
235	Q₈	—	332	2	no	—
236	Q₈	—	166	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	332	2	no	—
241	C₂×C₄	—	332	2	no	—
242	C₂×C₄	—	332	2	no	—
243	C₂×C₄	—	332	2	no	—
244	C₂×C₄	—	166	2	no	—
245	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
246	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
247	C₂×C₄	#14581754	1	1	yes	characteristic
248	C₂×C₄	#14581819	1	1	yes	characteristic
249	C₂×C₄	#14581819	1	1	yes	characteristic
250	C₂×C₄	#14581824	1	1	yes	characteristic
251	C₈	—	332	2	no	—
252	C₈	—	166	2	no	—
253	C₈	C₄²×D₈₃	1	1	yes	characteristic
254	C₈	#14581750	1	1	yes	characteristic
255	C₈	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
256	C₈	#14581754	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	#19127704	1	1	yes	characteristic
259	C₂²	#19127800	1	1	yes	characteristic
260	C₂²	#19127810	1	1	yes	characteristic
261	C₄	—	8	2	no	—
262	C₄	—	8	2	no	—
263	C₄	—	332	2	no	—
264	C₄	—	332	4	no	—
265	C₄	#19127548	1	1	yes	characteristic
266	C₄	#19127701	1	1	yes	characteristic
267	C₄	#19127804	1	1	yes	characteristic
268	C₄	#19127812	1	1	yes	characteristic
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	#24907287	1	1	yes	characteristic
272	C₂	#24909351	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19127807

C₂

yes

maximal, characteristic

C₂×#19127518

C₂

yes

maximal, characteristic

#24909182

C₂

yes

maximal, characteristic

C₂×#19127478

C₂

yes

maximal, characteristic

#24907349

C₂

yes

maximal, characteristic

#24905561

C₂

yes

maximal, characteristic

C₈₃×#1386

C₂

yes

maximal, characteristic

C₂²×Dic₃₃₂

C₂²

yes

characteristic

C₂×#14581757

—

C₂×#14581732

C₂²

yes

characteristic

#19127807

—

#19127807

C₄

yes

complemented

C₂×#14581728

C₂²

yes

—

—

C₂²

yes

characteristic

#19127518

C₄

yes

complemented

#19127504

C₄

yes

complemented

#19127478

—

#19127460

—

C₂×M₃₂×C₈₃

C₂²

yes

characteristic

#427×C₈₃

C₂²

yes

characteristic

C₂×#106×C₈₃

C₂²

yes

characteristic

C₂²×Dic₁₆₆

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

C₂×C₄

yes

—

C₂×Dic₃₃₂

C₂×C₄

yes

—

C₂×Dic₃₃₂

C₂×C₄

yes

—

C₂×#13050574

—

C₂²×C₈₃⋊C₈

D₄

yes

characteristic

#14581757

—

#14581757

—

C₂×Dic₈₃⋊C₄

—

C₂×C₄×Dic₈₃

—

C₂×C₄×Dic₈₃

D₄

yes

characteristic

#14581732

—

#14581732

—

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

—

#14581728

—

#14581728

C₂×C₄

yes

—

#14581715

—

#14581703

—

#14581702

C₂×C₄

yes

—

#14581702

D₄

yes

—

C₄×C₈₃⋊C₈

—

C₂²×C₈×C₈₃

C₂³

yes

characteristic

#117×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

M₃₂×C₈₃

—

M₃₂×C₈₃

C₂×C₄

yes

—

C₂×C₁₆×C₈₃

—

#107×C₈₃

C₂×C₄

yes

—

#106×C₈₃

—

#106×C₈₃

—

#106×C₈₃

C₂×C₄

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

C₄²

yes

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂²⋊C₄

yes

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₂²⋊C₄

yes

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

C₂×C₈×C₈₃

yes

C₂×C₈×C₈₃

yes

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₁₆×C₈₃

—

C₂³×C₈₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

100

C₂×C₄×C₈₃

C₄×D₄

yes

characteristic

101

C₂×C₄×C₈₃

C₄×D₄

yes

characteristic

102

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

characteristic

103

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

characteristic

104

C₂×C₄×C₈₃

#137

yes

characteristic

105

C₂×Dic₈₃

—

106

C₂×Dic₈₃

—

107

C₂×Dic₈₃

—

108

C₂×Dic₈₃

—

109

C₂×Dic₈₃

—

110

Dic₁₆₆

—

111

Dic₁₆₆

—

112

Dic₁₆₆

—

113

Dic₁₆₆

—

114

C₈×C₈₃

C₂×C₄²

yes

characteristic

115

C₈×C₈₃

#117

yes

characteristic

116

C₈×C₈₃

C₄×D₄

yes

characteristic

117

C₈×C₈₃

C₄×D₄

yes

characteristic

118

C₈₃⋊C₈

—

119

C₈₃⋊C₈

—

120

C₂²×C₈₃

—

121

C₂²×C₈₃

#418

yes

characteristic

122

C₂²×C₈₃

#440

yes

characteristic

123

C₂²×C₈₃

#442

yes

characteristic

124

C₄×C₈₃

—

125

C₄×C₈₃

—

126

C₄×C₈₃

C₄×C₂²⋊C₄

yes

characteristic

127

C₄×C₈₃

#417

yes

characteristic

128

C₄×C₈₃

#440

yes

characteristic

129

C₄×C₈₃

#442

yes

characteristic

130

Dic₈₃

—

131

Dic₈₃

—

132

#12648

—

maximal, Sylow

133

C₂×C₈₃

—

134

C₂×C₈₃

—

135

C₂×C₈₃

#1762

yes

characteristic

136

C₂×C₈₃

#2180

yes

—

137

C₂×#441

—

138

C₂×#362

—

139

#2141

—

140

C₂×#347

—

141

#1773

—

142

#1386

D₈₃

yes

characteristic

143

#1383

—

144

C₈₃

#12648

yes

characteristic, complemented, Sylow

145

C₂²×Q₁₆

—

146

C₂×C₄×Q₈

—

166

—

147

C₂×M₃₂

—

148

C₂×M₃₂

C₂×D₈₃

yes

characteristic

149

#441

—

150

#441

—

166

—

151

C₂×#108

—

152

#427

C₂×D₈₃

yes

characteristic

153

C₂×#106

C₂×D₈₃

yes

characteristic

154

C₂×#103

—

166

—

155

#405

—

166

—

156

C₂×#97

—

157

#362

—

158

#358

—

159

#347

—

166

—

160

#339

—

166

—

161

C₂²×Q₈

—

166

—

162

C₂×Q₁₆

—

163

C₂×Q₁₆

—

164

C₂×Q₁₆

—

165

C₂×Q₁₆

—

166

—

166

C₂×Q₁₆

—

166

—

167

C₂×Q₁₆

—

166

—

168

C₂×M₁₆

—

166

—

169

C₂²×C₈

—

170

C₂²×C₈

C₂²×D₈₃

yes

characteristic

171

C₄×Q₈

—

332

—

172

C₄×Q₈

—

166

—

173

#117

—

174

C₂×C₄⋊C₄

—

175

C₂×C₄⋊C₄

—

332

—

176

C₂×C₄²

—

332

—

177

C₂×C₄²

—

178

M₃₂

—

179

M₃₂

—

166

—

180

M₃₂

—

166

—

181

M₃₂

C₄×D₈₃

yes

—

182

C₂×C₁₆

—

183

C₂×C₁₆

—

184

#108

—

185

#108

—

166

—

186

#107

C₄×D₈₃

yes

—

187

#106

—

188

#106

—

189

#106

C₄×D₈₃

yes

—

190

#103

—

166

—

191

#97

—

332

—

192

#97

—

193

#97

—

194

C₄×C₈

—

332

—

195

C₂×Q₈

—

332

—

196

C₂×Q₈

—

332

—

197

C₂×Q₈

—

332

—

198

C₂×Q₈

—

166

—

199

C₂×Q₈

—

166

—

200

C₂×Q₈

—

166

—

201

C₂²×C₄

—

202

C₂²×C₄

—

332

—

203

C₂²×C₄

—

166

—

204

C₂²×C₄

D₄×D₈₃

yes

characteristic

205

Q₁₆

—

206

Q₁₆

—

166

—

207

Q₁₆

—

166

—

208

Q₁₆

—

166

—

209

M₁₆

—

332

—

210

M₁₆

—

166

—

211

C₂×C₈

—

332

—

212

C₂×C₈

—

213

C₂×C₈

—

166

—

214

C₂×C₈

—

166

—

215

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

216

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

217

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

218

C₂×C₈

D₄×D₈₃

yes

characteristic

219

C₂×C₈

#13050589

yes

characteristic

220

C₂×C₈

#13050591

yes

characteristic

221

C₄⋊C₄

—

222

C₄⋊C₄

—

223

C₄⋊C₄

—

224

C₄⋊C₄

—

332

—

225

C₂²⋊C₄

—

226

C₄²

—

227

C₄²

—

332

—

228

C₄²

—

229

C₄²

—

166

—

230

C₁₆

—

231

C₁₆

—

166

—

232

C₂³

#14581816

yes

characteristic

233

Q₈

—

332

—

234

Q₈

—

332

—

235

Q₈

—

332

—

236

Q₈

—

166

—

237

C₂×C₄

—

238

C₂×C₄

—

239

C₂×C₄

—

240

C₂×C₄

—

332

—

241

C₂×C₄

—

332

—

242

C₂×C₄

—

332

—

243

C₂×C₄

—

332

—

244

C₂×C₄

—

166

—

245

C₂×C₄

C₂²⋊C₄×D₈₃

yes

characteristic

246

C₂×C₄

(C₂²⋊D₈₃)⋊C₄

yes

characteristic

247

C₂×C₄

#14581754

yes

characteristic

248

C₂×C₄

#14581819

yes

characteristic

249

C₂×C₄

#14581819

yes

characteristic

250

C₂×C₄

#14581824

yes

characteristic

251

C₈

—

332

—

252

C₈

—

166

—

253

C₈

C₄²×D₈₃

yes

characteristic

254

C₈

#14581750

yes

characteristic

255

C₈

(C₂²⋊D₈₃)⋊C₄

yes

characteristic

256

C₈

#14581754

yes

characteristic

257

C₂²

—

258

C₂²

#19127704

yes

characteristic

259

C₂²

#19127800

yes

characteristic

260

C₂²

#19127810

yes

characteristic

261

C₄

—

262

C₄

—

263

C₄

—

332

—

264

C₄

—

332

—

265

C₄

#19127548

yes

characteristic

266

C₄

#19127701

yes

characteristic

267

C₄

#19127804

yes

characteristic

268

C₄

#19127812

yes

characteristic

269

C₂

—

270

C₂

—

271

C₂

#24907287

yes

characteristic

272

C₂

#24909351

yes

—

	subgroup	G/H
center	C₂²	#19127810
commutator	C₈×C₈₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1386	C₂
socle	C₂²×C₈₃	#442
prime radical	C₂³×C₈₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1386	D₈₃
hypocenter	C₈₃	#12648
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1386

subgroup

G/H

center

C₂²

#19127810

commutator

C₈×C₈₃

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#1386

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1386

N	H ≅ G/N	type
12. #19127807	C₄	semidirect
17. #19127518	C₄	semidirect
18. #19127504	C₄	semidirect
144. C₈₃	#12648	semidirect

H ≅ G/N

type

12. #19127807

C₄

semidirect

17. #19127518

C₄

semidirect

18. #19127504

C₄

semidirect

144. C₈₃

#12648

semidirect

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

subgroup	index
C₂×#19127807	2
C₂×#19127518	2
#24909182	2
C₂×#19127478	2
#24907349	2
#24905561	2
C₈₃×#1386	2
#12648	83

subgroup

index

C₈₃×#1386

Subgroups — #35319682

Conjugacy classes of subgroups

Complements

Maximal subgroups