Group #30749692 — subgroups

Subgroups — #30749692

Order 17152.

482 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 139 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	#22569233	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523960	C₂	1	1	yes	maximal, characteristic
3	#22558959	C₂	1	2	yes	maximal, complemented
4	#22558046	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1954	C₂	1	1	yes	maximal, characteristic
6	D₄×C₆₇⋊C₈	—	2	2	no	—
7	D₄×C₆₇⋊C₈	—	2	2	no	—
8	#16863449	C₂²	1	2	yes	—
9	C₂×#14523960	C₄	1	1	yes	characteristic
10	C₂×#14523960	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
12	C₄×#11761612	C₂²	1	2	yes	—
13	#16863165	C₄	1	2	yes	—
14	#16863148	—	2	2	no	—
15	#16863146	—	2	2	no	—
16	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#11761612	C₂×C₄	1	2	yes	—
21	C₂×#11761612	C₂×C₄	1	2	yes	—
22	C₂²×C₆₇⋊C₈	—	2	2	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
25	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
26	#14523960	—	2	2	no	—
27	#14523960	C₈	1	4	yes	complemented
28	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
29	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
30	#14523954	—	2	2	no	—
31	C₄×C₆₇⋊C₈	—	2	2	no	—
32	C₄×C₆₇⋊C₈	—	2	2	no	—
33	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	D₄	1	2	yes	—
35	C₂²×C₈×C₆₇	—	2	1	no	—
36	C₂²×C₈×C₆₇	—	2	1	no	—
37	C₄×D₄×C₆₇	—	2	2	no	—
38	C₄×D₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₆₇	Q₈	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₂²⋊C₈×C₆₇	—	4	4	no	—
44	C₄×C₈×C₆₇	—	2	2	no	—
45	#11761612	—	2	2	no	—
46	#11761612	—	2	2	no	—
47	#11761612	—	2	2	no	—
48	#11761612	C₂×C₈	1	4	yes	complemented
49	C₂×C₆₇⋊C₈	—	2	2	no	—
50	C₂×C₆₇⋊C₈	—	2	2	no	—
51	C₂×C₆₇⋊C₈	—	2	2	no	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	8	2	no	—
54	C₂×C₆₇⋊C₈	—	8	2	no	—
55	C₂⁴×C₆₇	C₄⋊C₄	1	2	yes	—
56	C₂×D₄×C₆₇	—	2	2	no	—
57	C₂×D₄×C₆₇	—	2	2	no	—
58	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₆₇	M₁₆	1	2	yes	—
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	4	2	no	—
63	C₂²×C₄×C₆₇	—	4	2	no	—
64	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	4	1	no	—
70	C₂×C₈×C₆₇	—	4	1	no	—
71	C₂×C₈×C₆₇	—	4	2	no	—
72	C₂×C₈×C₆₇	—	4	1	no	—
73	C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₂²⋊C₄×C₆₇	—	2	4	no	—
75	C₄²×C₆₇	C₂×D₄	1	2	yes	—
76	C₄²×C₆₇	#41	1	2	yes	—
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	C₄⋊C₈	1	4	yes	—
82	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	D₄×C₆₇	—	2	4	no	—
84	D₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	4	1	no	—
89	C₂×C₄×C₆₇	—	4	2	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	—	4	2	no	—
92	C₂×C₄×C₆₇	—	8	2	no	—
93	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
96	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#131	1	2	yes	—
100	C₈×C₆₇	—	4	2	no	—
101	C₈×C₆₇	—	8	1	no	—
102	C₆₇⋊C₈	—	2	4	no	—
103	C₆₇⋊C₈	—	4	2	no	—
104	C₆₇⋊C₈	—	4	2	no	—
105	C₆₇⋊C₈	—	4	2	no	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	—	2	1	no	—
108	C₂²×C₆₇	—	2	4	no	—
109	C₂²×C₆₇	—	4	4	no	—
110	C₂²×C₆₇	—	4	2	no	—
111	C₂²×C₆₇	—	4	2	no	—
112	C₂²×C₆₇	—	8	2	no	—
113	C₂²×C₆₇	#385	1	1	yes	characteristic
114	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₂²×C₆₇	#424	1	1	yes	characteristic
116	C₄×C₆₇	—	2	1	no	—
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	—	4	2	no	—
119	C₄×C₆₇	C₈×D₄	1	2	yes	—
120	C₄×C₆₇	#435	1	2	yes	—
121	C₄×C₆₇	#453	1	1	yes	characteristic
122	C₄×C₆₇	#455	1	1	yes	characteristic
123	#8835	—	67	1	no	maximal, Sylow
124	C₂×C₆₇	—	2	1	no	—
125	C₂×C₆₇	—	2	1	no	—
126	C₂×C₆₇	—	4	4	no	—
127	C₂×C₆₇	#1850	1	1	yes	characteristic
128	C₂×C₆₇	#1856	1	1	yes	characteristic
129	C₂×C₆₇	#1858	1	1	yes	characteristic
130	#2972	—	67	1	no	—
131	#1954	D₆₇	1	1	yes	characteristic
132	C₄×#100	—	67	1	no	—
133	#1565	—	67	2	no	—
134	#1278	—	67	2	no	—
135	C₆₇	#8835	1	1	yes	characteristic, complemented, Sylow
136	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
137	C₈×D₄	—	134	2	no	—
138	C₈×D₄	—	134	2	no	—
139	#431	—	67	2	no	—
140	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
141	C₂×#100	—	67	1	no	—
142	C₂×#100	—	67	1	no	—
143	C₂×C₂²⋊C₈	—	2	2	no	—
144	C₂×C₂²⋊C₈	—	134	2	no	—
145	C₄×M₁₆	—	67	2	no	—
146	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
147	#343	—	67	2	no	—
148	#335	—	134	2	no	—
149	#334	—	134	2	no	—
150	C₂²×D₄	C₄×D₆₇	1	1	yes	characteristic
151	C₂³×C₄	C₂²⋊D₆₇	1	2	yes	—
152	C₂×M₁₆	—	67	2	no	—
153	C₂×M₁₆	—	67	2	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	134	2	no	—
157	C₂²×C₈	—	134	2	no	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	2	2	no	—
160	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
161	C₂×C₂²⋊C₄	Dic₁₃₄	1	2	yes	—
162	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	67	2	no	—
165	C₄⋊C₈	—	67	2	no	—
166	#100	—	67	4	no	—
167	#100	—	134	2	no	—
168	C₂²⋊C₈	—	4	4	no	—
169	C₂²⋊C₈	—	134	4	no	—
170	C₂²⋊C₈	—	134	2	no	—
171	#97	—	134	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	134	2	no	—
174	C₄×C₈	—	134	2	no	—
175	C₂⁴	Dic₆₇⋊C₄	1	2	yes	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	C₈×D₆₇	1	2	yes	—
179	C₂×D₄	#11761611	1	2	yes	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
185	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
187	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
188	M₁₆	—	67	4	no	—
189	M₁₆	—	134	2	no	—
190	M₁₆	—	134	2	no	—
191	M₁₆	—	134	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	536	2	no	—
198	C₂×C₈	—	536	2	no	—
199	C₂×C₈	—	134	2	no	—
200	C₂×C₈	—	134	2	no	—
201	C₂×C₈	—	134	2	no	—
202	C₂×C₈	—	134	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₄²	D₄×D₆₇	1	2	yes	—
206	C₄²	#11761627	1	2	yes	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	—	4	2	no	—
211	C₂³	Dic₆₇⋊C₈	1	4	yes	—
212	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
213	D₄	—	2	4	no	—
214	D₄	—	2	4	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
224	C₂×C₄	#14524011	1	1	yes	characteristic
225	C₂×C₄	#14524020	1	2	yes	—
226	C₂×C₄	#14524024	1	1	yes	characteristic
227	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
228	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
229	C₂×C₄	#14524051	1	2	yes	—
230	C₈	—	4	2	no	—
231	C₈	—	8	1	no	—
232	C₈	—	268	2	no	—
233	C₈	—	268	2	no	—
234	C₈	—	268	2	no	—
235	C₈	—	134	4	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	4	4	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	#16863268	1	1	yes	characteristic
244	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
245	C₂²	#16863424	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	2	no	—
248	C₄	—	4	2	no	—
249	C₄	(C₂²⋊D₆₇)⋊C₈	1	2	yes	—
250	C₄	#16863464	1	2	yes	—
251	C₄	#16863563	1	1	yes	characteristic
252	C₄	#16863573	1	1	yes	characteristic
253	C₂	—	2	1	no	—
254	C₂	—	2	1	no	—
255	C₂	—	4	4	no	—
256	C₂	#22560386	1	1	yes	characteristic
257	C₂	#22560418	1	1	yes	characteristic
258	C₂	#22560428	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22569233

C₂

yes

maximal, characteristic

C₄×#14523960

C₂

yes

maximal, characteristic

#22558959

C₂

yes

maximal, complemented

#22558046

C₂

yes

maximal, complemented

C₆₇×#1954

C₂

yes

maximal, characteristic

D₄×C₆₇⋊C₈

—

D₄×C₆₇⋊C₈

—

#16863449

C₂²

yes

—

C₂×#14523960

C₄

yes

characteristic

C₂×#14523960

C₄

yes

characteristic

C₂×C₂²⋊(C₆₇⋊C₈)

—

C₄×#11761612

C₂²

yes

—

#16863165

C₄

yes

—

#16863148

—

#16863146

—

C₂×C₄×D₄×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×#11761612

C₂×C₄

yes

—

C₂×#11761612

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

#14523960

—

#14523960

C₈

yes

complemented

C₂²⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×D₄×C₆₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

—

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

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

—

—

—

C₂×C₈

yes

complemented

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×C₈

yes

—

C₂×D₄×C₆₇

M₁₆

yes

—

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

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

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

C₄⋊C₈

yes

—

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

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₇

#131

yes

—

100

C₈×C₆₇

—

101

C₈×C₆₇

—

102

C₆₇⋊C₈

—

103

C₆₇⋊C₈

—

104

C₆₇⋊C₈

—

105

C₆₇⋊C₈

—

106

C₂²×C₆₇

—

107

C₂²×C₆₇

—

108

C₂²×C₆₇

—

109

C₂²×C₆₇

—

110

C₂²×C₆₇

—

111

C₂²×C₆₇

—

112

C₂²×C₆₇

—

113

C₂²×C₆₇

#385

yes

characteristic

114

C₂²×C₆₇

C₂×C₄⋊C₈

yes

characteristic

115

C₂²×C₆₇

#424

yes

characteristic

116

C₄×C₆₇

—

117

C₄×C₆₇

—

118

C₄×C₆₇

—

119

C₄×C₆₇

C₈×D₄

yes

—

120

C₄×C₆₇

#435

yes

—

121

C₄×C₆₇

#453

yes

characteristic

122

C₄×C₆₇

yes

123

—

124

C₂×C₆₇

—

125

C₂×C₆₇

—

126

C₂×C₆₇

—

127

C₂×C₆₇

#1850

yes

characteristic

128

C₂×C₆₇

#1856

yes

characteristic

129

C₂×C₆₇

#1858

yes

characteristic

130

#2972

—

131

#1954

D₆₇

yes

characteristic

132

C₄×#100

—

133

#1565

—

134

#1278

—

135

C₆₇

#8835

yes

characteristic, complemented, Sylow

136

C₂×C₄×D₄

C₂×D₆₇

yes

characteristic

137

C₈×D₄

—

134

—

138

C₈×D₄

—

134

—

139

#431

—

140

C₂×C₄⋊C₈

C₂×D₆₇

yes

characteristic

141

C₂×#100

—

142

C₂×#100

—

143

C₂×C₂²⋊C₈

—

144

C₂×C₂²⋊C₈

—

134

—

145

C₄×M₁₆

—

146

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

147

#343

—

148

#335

—

134

—

149

#334

—

134

—

150

C₂²×D₄

C₄×D₆₇

yes

characteristic

151

C₂³×C₄

C₂²⋊D₆₇

yes

—

152

C₂×M₁₆

—

153

C₂×M₁₆

—

154

C₂²×C₈

—

155

C₂²×C₈

—

156

C₂²×C₈

—

134

—

157

C₂²×C₈

—

134

—

158

C₄×D₄

—

159

C₄×D₄

—

160

C₂×C₄⋊C₄

C₄×D₆₇

yes

characteristic

161

C₂×C₂²⋊C₄

Dic₁₃₄

yes

—

162

C₂×C₄²

C₂²×D₆₇

yes

characteristic

163

C₄⋊C₈

—

164

C₄⋊C₈

—

165

C₄⋊C₈

—

166

#100

—

167

#100

—

134

—

168

C₂²⋊C₈

—

169

C₂²⋊C₈

—

134

—

170

C₂²⋊C₈

—

134

—

171

#97

—

134

—

172

C₄×C₈

—

173

C₄×C₈

—

134

—

174

C₄×C₈

—

134

—

175

C₂⁴

Dic₆₇⋊C₄

yes

—

176

C₂×D₄

—

177

C₂×D₄

—

178

C₂×D₄

C₈×D₆₇

yes

—

179

C₂×D₄

#11761611

yes

—

180

C₂²×C₄

—

181

C₂²×C₄

—

182

C₂²×C₄

—

183

C₂²×C₄

—

184

C₂²×C₄

Dic₆₇⋊C₄

yes

—

185

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

186

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

187

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

188

M₁₆

—

189

M₁₆

—

134

—

190

M₁₆

—

134

—

191

M₁₆

—

134

—

192

C₂×C₈

—

193

C₂×C₈

—

194

C₂×C₈

—

195

C₂×C₈

—

196

C₂×C₈

—

197

C₂×C₈

—

536

—

198

C₂×C₈

—

536

—

199

C₂×C₈

—

134

—

200

C₂×C₈

—

134

—

201

C₂×C₈

—

134

—

202

C₂×C₈

—

134

—

203

C₄⋊C₄

—

204

C₂²⋊C₄

—

205

C₄²

D₄×D₆₇

yes

—

206

C₄²

#11761627

yes

—

207

C₂³

—

208

C₂³

—

209

C₂³

—

210

C₂³

—

211

C₂³

Dic₆₇⋊C₈

yes

—

212

C₂³

C₂×Dic₆₇⋊C₄

yes

characteristic

213

D₄

—

214

D₄

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

—

222

C₂×C₄

—

223

C₂×C₄

(C₂²⋊D₆₇)⋊C₄

yes

—

224

C₂×C₄

#14524011

yes

characteristic

225

C₂×C₄

#14524020

yes

—

226

C₂×C₄

#14524024

yes

characteristic

227

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

228

C₂×C₄

C₂×#11761611

yes

characteristic

229

C₂×C₄

#14524051

yes

—

230

C₈

—

231

C₈

—

232

C₈

—

268

—

233

C₈

—

268

—

234

C₈

—

268

—

235

C₈

—

134

—

236

C₂²

—

237

C₂²

—

238

C₂²

—

239

C₂²

—

240

C₂²

—

241

C₂²

—

242

C₂²

—

243

C₂²

#16863268

yes

characteristic

244

C₂²

C₂×Dic₆₇⋊C₈

yes

characteristic

245

C₂²

#16863424

yes

characteristic

246

C₄

—

247

C₄

—

248

C₄

—

249

C₄

(C₂²⋊D₆₇)⋊C₈

yes

—

250

C₄

#16863464

yes

—

251

C₄

#16863563

yes

characteristic

252

C₄

#16863573

yes

characteristic

253

C₂

—

254

C₂

—

255

C₂

—

256

C₂

#22560386

yes

characteristic

257

C₂

#22560418

yes

characteristic

258

C₂

#22560428

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524011
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1954	C₂
socle	C₂²×C₆₇	#385
prime radical	C₂²×D₄×C₆₇	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1954	D₆₇
hypocenter	C₆₇	#8835
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1954

subgroup

G/H

center

C₂×C₄

#14524011

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1954

C₂

socle

C₂²×C₆₇

#385

prime radical

C₂²×D₄×C₆₇

C₂×C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1954

N	H ≅ G/N	type
3. #22558959	C₂	semidirect
4. #22558046	C₂	semidirect
27. #14523960	C₈	semidirect
48. #11761612	C₂×C₈	semidirect
135. C₆₇	#8835	semidirect

H ≅ G/N

type

3. #22558959

C₂

semidirect

4. #22558046

C₂

semidirect

27. #14523960

C₈

semidirect

48. #11761612

C₂×C₈

semidirect

135. C₆₇

#8835

semidirect

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

subgroup	index
#22569233	2
C₄×#14523960	2
#22558959	2
#22558046	2
C₆₇×#1954	2
#8835	67

subgroup

index

C₆₇×#1954

Subgroups — #30749692

Conjugacy classes of subgroups

Complements

Maximal subgroups