Group #30749705 — subgroups

Subgroups — #30749705

Order 17152.

454 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22569250	C₂	1	1	yes	maximal, characteristic
2	#22561004	C₂	1	1	yes	maximal, characteristic
3	#22558961	C₂	1	2	yes	maximal, complemented
4	#22558047	C₂	1	2	yes	maximal, complemented
5	C₄×#100×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863474	—	2	2	no	—
7	#16863474	—	2	2	no	—
8	#16863449	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
10	#16863360	—	2	2	no	—
11	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#16863149	—	2	2	no	—
14	#16863147	—	2	2	no	—
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#100×C₆₇	C₄	1	2	yes	—
17	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
18	#343×C₆₇	C₄	1	1	yes	characteristic
19	#343×C₆₇	C₄	1	1	yes	characteristic
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂²×C₆₇⋊C₈	—	2	1	no	—
23	C₂²×C₆₇⋊C₈	—	2	1	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
26	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
27	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
28	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₂²⋊(C₆₇⋊C₈)	—	4	4	no	—
30	C₄×C₆₇⋊C₈	—	2	2	no	—
31	C₄×C₆₇⋊C₈	—	2	2	no	—
32	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	D₄	1	2	yes	—
34	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
36	C₄×D₄×C₆₇	—	2	2	no	—
37	C₄×D₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₆₇	Q₈	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	#100×C₆₇	—	2	4	no	—
42	#97×C₆₇	—	2	2	no	—
43	C₄×C₈×C₆₇	—	2	2	no	—
44	#11761612	—	8	2	no	—
45	#11761612	—	8	2	no	—
46	C₂×C₆₇⋊C₈	—	2	2	no	—
47	C₂×C₆₇⋊C₈	—	2	2	no	—
48	C₂×C₆₇⋊C₈	—	2	2	no	—
49	C₂×C₆₇⋊C₈	—	4	1	no	—
50	C₂×C₆₇⋊C₈	—	4	1	no	—
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₂×C₆₇⋊C₈	—	4	1	no	—
53	C₂⁴×C₆₇	C₄⋊C₄	1	2	yes	—
54	C₂×D₄×C₆₇	—	2	2	no	—
55	C₂×D₄×C₆₇	—	2	2	no	—
56	C₂×D₄×C₆₇	—	2	1	no	—
57	C₂×D₄×C₆₇	—	2	1	no	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	—	4	2	no	—
61	C₂²×C₄×C₆₇	—	4	2	no	—
62	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
66	M₁₆×C₆₇	—	2	2	no	—
67	M₁₆×C₆₇	—	2	2	no	—
68	M₁₆×C₆₇	—	2	2	no	—
69	M₁₆×C₆₇	M₁₆	1	4	yes	complemented
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₂²⋊C₄×C₆₇	—	2	4	no	—
74	C₄²×C₆₇	C₂×D₄	1	2	yes	—
75	C₄²×C₆₇	#41	1	2	yes	—
76	C₂³×C₆₇	—	2	2	no	—
77	C₂³×C₆₇	—	2	2	no	—
78	C₂³×C₆₇	—	2	2	no	—
79	C₂³×C₆₇	—	4	2	no	—
80	C₂³×C₆₇	—	4	2	no	—
81	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	D₄×C₆₇	—	4	2	no	—
83	D₄×C₆₇	—	4	2	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	4	1	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	—	4	2	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	—	8	2	no	—
92	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
95	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
99	C₈×C₆₇	—	2	4	no	—
100	C₈×C₆₇	—	4	2	no	—
101	C₆₇⋊C₈	—	4	2	no	—
102	C₆₇⋊C₈	—	4	2	no	—
103	C₆₇⋊C₈	—	4	2	no	—
104	C₆₇⋊C₈	—	8	1	no	—
105	C₂²×C₆₇	—	2	1	no	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	—	4	4	no	—
108	C₂²×C₆₇	—	4	2	no	—
109	C₂²×C₆₇	—	4	2	no	—
110	C₂²×C₆₇	—	4	2	no	—
111	C₂²×C₆₇	—	8	2	no	—
112	C₂²×C₆₇	#385	1	1	yes	characteristic
113	C₂²×C₆₇	#423	1	1	yes	characteristic
114	C₂²×C₆₇	#424	1	1	yes	characteristic
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	—	4	2	no	—
118	C₄×C₆₇	#435	1	2	yes	—
119	C₄×C₆₇	#436	1	2	yes	—
120	C₄×C₆₇	#453	1	1	yes	characteristic
121	C₄×C₆₇	#455	1	1	yes	characteristic
122	#8837	—	67	1	no	maximal, Sylow
123	C₂×C₆₇	—	2	1	no	—
124	C₂×C₆₇	—	2	1	no	—
125	C₂×C₆₇	—	8	2	no	—
126	C₂×C₆₇	#1851	1	1	yes	characteristic
127	C₂×C₆₇	#1856	1	1	yes	characteristic
128	C₂×C₆₇	#1858	1	1	yes	characteristic
129	#2975	—	67	1	no	—
130	#1954	—	67	1	no	—
131	C₄×#100	D₆₇	1	1	yes	characteristic
132	#1565	—	67	2	no	—
133	#1278	—	67	2	no	—
134	C₆₇	#8837	1	1	yes	characteristic, complemented, Sylow
135	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
136	#436	—	134	2	no	—
137	#436	—	134	2	no	—
138	#431	—	67	2	no	—
139	C₂×C₄⋊C₈	—	67	1	no	—
140	C₂×#100	Dic₆₇	1	2	yes	—
141	#407	—	134	2	no	—
142	C₂×C₂²⋊C₈	—	134	2	no	—
143	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
144	C₂×C₄×C₈	—	67	1	no	—
145	#343	Dic₆₇	1	1	yes	characteristic
146	#343	Dic₆₇	1	1	yes	characteristic
147	#335	—	134	2	no	—
148	#334	—	134	2	no	—
149	C₂²×D₄	C₂×Dic₆₇	1	1	yes	characteristic
150	C₂³×C₄	D₂₆₈	1	2	yes	—
151	C₂×M₁₆	—	134	2	no	—
152	C₂×M₁₆	—	134	2	no	—
153	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
154	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
155	C₂²×C₈	—	134	1	no	—
156	C₂²×C₈	—	134	1	no	—
157	C₄×D₄	—	2	2	no	—
158	C₄×D₄	—	2	2	no	—
159	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	Dic₁₃₄	1	2	yes	—
161	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
162	C₄⋊C₈	—	67	2	no	—
163	C₄⋊C₈	—	67	2	no	—
164	C₄⋊C₈	—	134	2	no	—
165	#100	—	2	4	no	—
166	C₂²⋊C₈	—	268	4	no	—
167	C₂²⋊C₈	—	134	4	no	—
168	C₂²⋊C₈	—	134	2	no	—
169	#97	—	2	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	—	134	2	no	—
172	C₄×C₈	—	134	2	no	—
173	C₂⁴	C₄⋊Dic₆₇	1	2	yes	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	2	1	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
183	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	2	2	no	—
188	M₁₆	—	2	2	no	—
189	M₁₆	—	536	2	no	—
190	M₁₆	—	536	2	no	—
191	M₁₆	#11761612	1	4	yes	complemented
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	268	1	no	—
195	C₂×C₈	—	268	1	no	—
196	C₂×C₈	—	268	2	no	—
197	C₂×C₈	—	268	1	no	—
198	C₂×C₈	—	134	2	no	—
199	C₂×C₈	—	134	2	no	—
200	C₂×C₈	—	134	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₄²	D₄×D₆₇	1	2	yes	—
204	C₄²	#11761627	1	2	yes	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	4	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
211	D₄	—	4	2	no	—
212	D₄	—	4	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
222	C₂×C₄	#14524013	1	1	yes	characteristic
223	C₂×C₄	#14524020	1	2	yes	—
224	C₂×C₄	#14524027	1	1	yes	characteristic
225	C₂×C₄	C₂×#11761612	1	2	yes	—
226	C₂×C₄	#14524050	1	1	yes	characteristic
227	C₂×C₄	#14524050	1	1	yes	characteristic
228	C₈	—	2	4	no	—
229	C₈	—	4	2	no	—
230	C₈	—	268	2	no	—
231	C₈	—	268	2	no	—
232	C₈	—	268	2	no	—
233	C₈	—	536	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	4	4	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	8	2	no	—
241	C₂²	#16863271	1	1	yes	characteristic
242	C₂²	#16863420	1	1	yes	characteristic
243	C₂²	#16863423	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	4	2	no	—
247	C₄	#16863468	1	2	yes	—
248	C₄	#16863474	1	2	yes	—
249	C₄	#16863566	1	1	yes	characteristic
250	C₄	#16863576	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	—	8	2	no	—
254	C₂	#22560394	1	1	yes	characteristic
255	C₂	#22560421	1	1	yes	characteristic
256	C₂	#22560431	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22569250

C₂

yes

maximal, characteristic

#22561004

C₂

yes

maximal, characteristic

#22558961

C₂

yes

maximal, complemented

#22558047

C₂

yes

maximal, complemented

C₄×#100×C₆₇

C₂

yes

maximal, characteristic

#16863474

—

#16863474

—

#16863449

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

#16863360

—

C₂×C₂²⋊(C₆₇⋊C₈)

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

#16863149

—

#16863147

—

C₂×C₄×D₄×C₆₇

C₂²

yes

characteristic

C₂×#100×C₆₇

C₄

yes

—

C₄×M₁₆×C₆₇

C₂²

yes

—

#343×C₆₇

C₄

yes

characteristic

#343×C₆₇

C₄

yes

characteristic

C₂×#11761612

—

C₂×#11761612

—

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×D₄×C₆₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₆₇

D₄

yes

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₄×D₄×C₆₇

—

C₄×D₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₂²⋊C₄×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#100×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂⁴×C₆₇

C₄⋊C₄

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

M₁₆

yes

complemented

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×C₄⋊C₄

yes

characteristic

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#120

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

#123

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₈×C₆₇

—

100

C₈×C₆₇

—

101

C₆₇⋊C₈

—

102

C₆₇⋊C₈

—

103

C₆₇⋊C₈

—

104

C₆₇⋊C₈

—

105

C₂²×C₆₇

—

106

C₂²×C₆₇

—

107

C₂²×C₆₇

—

108

C₂²×C₆₇

—

109

C₂²×C₆₇

—

110

C₂²×C₆₇

—

111

C₂²×C₆₇

—

112

C₂²×C₆₇

#385

yes

characteristic

113

C₂²×C₆₇

#423

yes

characteristic

114

C₂²×C₆₇

#424

yes

characteristic

115

C₄×C₆₇

—

116

C₄×C₆₇

—

117

C₄×C₆₇

—

118

C₄×C₆₇

#435

yes

—

119

C₄×C₆₇

#436

yes

—

120

C₄×C₆₇

#453

yes

characteristic

121

C₄×C₆₇

yes

122

—

123

C₂×C₆₇

—

124

C₂×C₆₇

—

125

C₂×C₆₇

—

126

C₂×C₆₇

#1851

yes

characteristic

127

C₂×C₆₇

#1856

yes

characteristic

128

C₂×C₆₇

#1858

yes

characteristic

129

#2975

—

130

#1954

—

131

C₄×#100

D₆₇

yes

characteristic

132

#1565

—

133

#1278

—

134

C₆₇

#8837

yes

characteristic, complemented, Sylow

135

C₂×C₄×D₄

C₂×D₆₇

yes

characteristic

136

#436

—

134

—

137

#436

—

134

—

138

#431

—

139

C₂×C₄⋊C₈

—

140

C₂×#100

Dic₆₇

yes

—

141

#407

—

134

—

142

C₂×C₂²⋊C₈

—

134

—

143

C₄×M₁₆

C₂×D₆₇

yes

—

144

C₂×C₄×C₈

—

145

#343

Dic₆₇

yes

characteristic

146

#343

Dic₆₇

yes

characteristic

147

#335

—

134

—

148

#334

—

134

—

149

C₂²×D₄

C₂×Dic₆₇

yes

characteristic

150

C₂³×C₄

D₂₆₈

yes

—

151

C₂×M₁₆

—

134

—

152

C₂×M₁₆

—

134

—

153

C₂×M₁₆

C₂×Dic₆₇

yes

—

154

C₂×M₁₆

C₂×Dic₆₇

yes

—

155

C₂²×C₈

—

134

—

156

C₂²×C₈

—

134

—

157

C₄×D₄

—

158

C₄×D₄

—

159

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

characteristic

160

C₂×C₂²⋊C₄

Dic₁₃₄

yes

—

161

C₂×C₄²

C₂²×D₆₇

yes

characteristic

162

C₄⋊C₈

—

163

C₄⋊C₈

—

164

C₄⋊C₈

—

134

—

165

#100

—

166

C₂²⋊C₈

—

268

—

167

C₂²⋊C₈

—

134

—

168

C₂²⋊C₈

—

134

—

169

#97

—

170

C₄×C₈

—

171

C₄×C₈

—

134

—

172

C₄×C₈

—

134

—

173

C₂⁴

C₄⋊Dic₆₇

yes

—

174

C₂×D₄

—

175

C₂×D₄

—

176

C₂×D₄

—

177

C₂×D₄

—

178

C₂²×C₄

—

179

C₂²×C₄

—

180

C₂²×C₄

—

181

C₂²×C₄

—

182

C₂²×C₄

C₄⋊Dic₆₇

yes

—

183

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

184

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

185

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

186

M₁₆

—

187

M₁₆

—

188

M₁₆

—

189

M₁₆

—

536

—

190

M₁₆

—

536

—

191

M₁₆

#11761612

yes

complemented

192

C₂×C₈

—

193

C₂×C₈

—

194

C₂×C₈

—

268

—

195

C₂×C₈

—

268

—

196

C₂×C₈

—

268

—

197

C₂×C₈

—

268

—

198

C₂×C₈

—

134

—

199

C₂×C₈

—

134

—

200

C₂×C₈

—

134

—

201

C₄⋊C₄

—

202

C₂²⋊C₄

—

203

C₄²

D₄×D₆₇

yes

—

204

C₄²

#11761627

yes

—

205

C₂³

—

206

C₂³

—

207

C₂³

—

208

C₂³

—

209

C₂³

—

210

C₂³

C₂×C₄⋊Dic₆₇

yes

characteristic

211

D₄

—

212

D₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

D₄×Dic₆₇

yes

—

222

C₂×C₄

#14524013

yes

characteristic

223

C₂×C₄

#14524020

yes

—

224

C₂×C₄

#14524027

yes

characteristic

225

C₂×C₄

C₂×#11761612

yes

—

226

C₂×C₄

#14524050

yes

characteristic

227

C₂×C₄

#14524050

yes

characteristic

228

C₈

—

229

C₈

—

230

C₈

—

268

—

231

C₈

—

268

—

232

C₈

—

268

—

233

C₈

—

536

—

234

C₂²

—

235

C₂²

—

236

C₂²

—

237

C₂²

—

238

C₂²

—

239

C₂²

—

240

C₂²

—

241

C₂²

#16863271

yes

characteristic

242

C₂²

#16863420

yes

characteristic

243

C₂²

#16863423

yes

characteristic

244

C₄

—

245

C₄

—

246

C₄

—

247

C₄

#16863468

yes

—

248

C₄

#16863474

yes

—

249

C₄

#16863566

yes

characteristic

250

C₄

#16863576

yes

characteristic

251

C₂

—

252

C₂

—

253

C₂

—

254

C₂

#22560394

yes

characteristic

255

C₂

#22560421

yes

characteristic

256

C₂

#22560431

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524013
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#100×C₆₇	C₂
socle	C₂²×C₆₇	#385
prime radical	C₂²×D₄×C₆₇	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#100	D₆₇
hypocenter	C₆₇	#8837
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#100

subgroup

G/H

center

C₂×C₄

#14524013

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₄×#100×C₆₇

C₂

socle

C₂²×C₆₇

#385

prime radical

C₂²×D₄×C₆₇

C₂×C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#100

N	H ≅ G/N	type
3. #22558961	C₂	semidirect
4. #22558047	C₂	semidirect
69. M₁₆×C₆₇	M₁₆	semidirect
134. C₆₇	#8837	semidirect
191. M₁₆	#11761612	semidirect

H ≅ G/N

type

3. #22558961

C₂

semidirect

4. #22558047

C₂

semidirect

69. M₁₆×C₆₇

M₁₆

semidirect

134. C₆₇

#8837

semidirect

191. M₁₆

#11761612

semidirect

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

subgroup	index
#22569250	2
#22561004	2
#22558961	2
#22558047	2
C₄×#100×C₆₇	2
#8837	67

subgroup

index

C₄×#100×C₆₇

Subgroups — #30749705

Conjugacy classes of subgroups

Complements

Maximal subgroups