Group #30754904 — subgroups

Subgroups — #30754904

Order 17152.

478 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 159 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	#22560326	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863142	C₂	1	2	yes	maximal
4	C₂×#16863142	C₂	1	2	yes	maximal
5	C₆₇×#2862	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
9	#16863151	—	2	1	no	—
10	#16863151	—	2	2	no	—
11	#16863142	—	2	4	no	—
12	#16863142	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	#391×C₆₇	—	2	1	no	—
16	#391×C₆₇	—	2	1	no	—
17	#391×C₆₇	—	2	2	no	—
18	#382×C₆₇	C₄	1	4	yes	—
19	C₄³×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₂²×C₆₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
26	C₄×C₆₇⋊C₈	—	4	2	no	—
27	C₂²×Q₈×C₆₇	—	2	1	no	—
28	C₂²×Q₈×C₆₇	—	2	1	no	—
29	C₄⋊Q₈×C₆₇	—	2	2	no	—
30	C₄⋊Q₈×C₆₇	—	2	2	no	—
31	C₄⋊Q₈×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	#95×C₆₇	—	2	4	no	—
43	#95×C₆₇	—	2	4	no	—
44	C₂×C₆₇⋊C₈	—	2	4	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×C₆₇⋊C₈	—	4	4	no	—
49	C₂×Q₈×C₆₇	—	4	2	no	—
50	C₂×Q₈×C₆₇	—	4	1	no	—
51	C₂×Q₈×C₆₇	—	4	1	no	—
52	C₂×Q₈×C₆₇	—	4	2	no	—
53	C₂×Q₈×C₆₇	—	4	4	no	—
54	C₂×Q₈×C₆₇	—	4	1	no	—
55	C₂×Q₈×C₆₇	—	4	1	no	—
56	C₂²×C₄×C₆₇	—	2	2	no	—
57	C₂²×C₄×C₆₇	—	2	2	no	—
58	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₆₇	—	2	4	no	—
64	C₄⋊C₄×C₆₇	—	2	4	no	—
65	C₄⋊C₄×C₆₇	—	4	2	no	—
66	C₄⋊C₄×C₆₇	—	4	1	no	—
67	C₄⋊C₄×C₆₇	—	4	1	no	—
68	C₄²×C₆₇	—	2	4	no	—
69	C₄²×C₆₇	—	2	4	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	2	1	no	—
72	C₄²×C₆₇	—	2	1	no	—
73	C₄²×C₆₇	C₂×D₄	1	2	yes	—
74	C₄²×C₆₇	#41	1	2	yes	—
75	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Q₈×C₆₇	—	8	2	no	—
77	Q₈×C₆₇	—	8	2	no	—
78	Q₈×C₆₇	—	8	2	no	—
79	Q₈×C₆₇	—	8	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	4	2	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	4	no	—
89	C₂×C₄×C₆₇	—	4	2	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	#101	1	4	yes	—
92	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₆₇	#121	1	2	yes	—
94	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#135	1	2	yes	—
97	C₂×C₄×C₆₇	#135	1	4	yes	—
98	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
100	C₆₇⋊C₈	—	4	4	no	—
101	C₆₇⋊C₈	—	8	2	no	—
102	C₂²×C₆₇	#390	1	1	yes	characteristic
103	C₂²×C₆₇	C₂×#101	1	2	yes	—
104	C₂²×C₆₇	#416	1	1	yes	characteristic
105	C₂²×C₆₇	#416	1	2	yes	—
106	C₂²×C₆₇	#417	1	1	yes	characteristic
107	C₄×C₆₇	—	2	4	no	—
108	C₄×C₆₇	—	2	2	no	—
109	C₄×C₆₇	—	4	2	no	—
110	C₄×C₆₇	—	8	2	no	—
111	C₄×C₆₇	—	8	2	no	—
112	C₄×C₆₇	#463	1	2	yes	—
113	C₄×C₆₇	#464	1	2	yes	—
114	C₄×C₆₇	#488	1	2	yes	—
115	C₄×C₆₇	#495	1	2	yes	—
116	#9700	—	67	1	no	maximal, Sylow
117	C₂×C₆₇	#1544	1	2	yes	—
118	C₂×C₆₇	#1552	1	2	yes	—
119	C₂×C₆₇	#1958	1	1	yes	characteristic
120	C₂×C₆₇	#1966	1	1	yes	characteristic
121	C₂×C₆₇	#1983	1	1	yes	characteristic
122	#2862	D₆₇	1	1	yes	characteristic
123	#1831	—	67	1	no	—
124	C₄×C₄⋊C₈	—	67	1	no	—
125	C₂×#332	—	67	2	no	—
126	C₂×#332	—	67	2	no	—
127	C₆₇	#9700	1	1	yes	characteristic, complemented, Sylow
128	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
129	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
130	C₂×C₄⋊C₈	—	67	2	no	—
131	C₂×C₄⋊C₈	—	67	2	no	—
132	C₂×C₄×C₈	—	134	1	no	—
133	#391	—	2	1	no	—
134	#391	—	2	2	no	—
135	#391	—	2	1	no	—
136	#382	Dic₆₇	1	4	yes	—
137	C₄³	C₂×D₆₇	1	1	yes	characteristic
138	#336	—	134	2	no	—
139	#336	—	134	1	no	—
140	#332	—	134	4	no	—
141	#332	—	134	4	no	—
142	C₂²×Q₈	—	2	1	no	—
143	C₂²×Q₈	—	2	1	no	—
144	C₂²×C₈	—	134	2	no	—
145	C₂²×C₈	—	134	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
154	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
156	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
159	C₄⋊C₈	—	67	4	no	—
160	C₄⋊C₈	—	134	2	no	—
161	C₄⋊C₈	—	134	2	no	—
162	C₄⋊C₈	—	134	2	no	—
163	C₄×C₈	—	268	2	no	—
164	#95	—	2	4	no	—
165	#95	—	2	4	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂×Q₈	—	4	1	no	—
169	C₂×Q₈	—	4	4	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
176	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
177	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
180	C₂×C₈	—	268	2	no	—
181	C₂×C₈	—	268	2	no	—
182	C₂×C₈	—	268	2	no	—
183	C₂×C₈	—	268	4	no	—
184	C₂×C₈	—	134	4	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₄⋊C₄	—	4	1	no	—
190	C₄²	—	2	4	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	D₄×D₆₇	1	2	yes	—
196	C₄²	#11761627	1	2	yes	—
197	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
198	Q₈	—	8	2	no	—
199	Q₈	—	8	2	no	—
200	Q₈	—	8	2	no	—
201	Q₈	—	8	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	#14523962	1	4	yes	—
214	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
215	C₂×C₄	#14524014	1	2	yes	—
216	C₂×C₄	#14524030	1	1	yes	characteristic
217	C₂×C₄	#14524039	1	1	yes	characteristic
218	C₂×C₄	#14524063	1	4	yes	—
219	C₂×C₄	#14524063	1	2	yes	—
220	C₂×C₄	#14524068	1	1	yes	characteristic
221	C₂×C₄	#14524073	1	1	yes	characteristic
222	C₈	—	268	4	no	—
223	C₈	—	536	2	no	—
224	C₂²	#16863299	1	1	yes	characteristic
225	C₂²	C₂×#14523962	1	2	yes	—
226	C₂²	#16863392	1	1	yes	characteristic
227	C₂²	#16863392	1	2	yes	—
228	C₂²	#16863397	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	4	2	no	—
232	C₄	—	8	2	no	—
233	C₄	—	8	2	no	—
234	C₄	#16863608	1	2	yes	—
235	C₄	#16863615	1	2	yes	—
236	C₄	#16863750	1	2	yes	—
237	C₄	#16863778	1	2	yes	—
238	C₂	#22558874	1	2	yes	—
239	C₂	#22558909	1	2	yes	—
240	C₂	#22561021	1	1	yes	characteristic
241	C₂	#22561054	1	1	yes	characteristic
242	C₂	#22561142	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560326

C₂

yes

maximal, characteristic

C₄×C₄⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×#16863142

C₂

yes

maximal

C₂×#16863142

C₂

yes

maximal

C₆₇×#2862

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

—

—

—

—

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

#391×C₆₇

—

#391×C₆₇

—

#391×C₆₇

—

#382×C₆₇

C₄

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₄×C₆₇⋊C₈

—

C₂²×Q₈×C₆₇

—

C₂²×Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#95×C₆₇

—

#95×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×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

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#101

yes

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

100

C₆₇⋊C₈

—

101

C₆₇⋊C₈

—

102

C₂²×C₆₇

#390

yes

characteristic

103

C₂²×C₆₇

C₂×#101

yes

—

104

C₂²×C₆₇

#416

yes

characteristic

105

C₂²×C₆₇

#416

yes

—

106

C₂²×C₆₇

#417

yes

characteristic

107

C₄×C₆₇

—

108

C₄×C₆₇

—

109

C₄×C₆₇

—

110

C₄×C₆₇

—

111

C₄×C₆₇

—

112

C₄×C₆₇

#463

yes

—

113

C₄×C₆₇

#464

yes

—

114

C₄×C₆₇

#488

yes

—

115

C₄×C₆₇

#495

yes

—

116

#9700

—

maximal, Sylow

117

C₂×C₆₇

#1544

yes

—

118

C₂×C₆₇

#1552

yes

—

119

C₂×C₆₇

#1958

yes

characteristic

120

C₂×C₆₇

#1966

yes

characteristic

121

C₂×C₆₇

#1983

yes

characteristic

122

#2862

D₆₇

yes

characteristic

123

#1831

—

124

C₄×C₄⋊C₈

—

125

C₂×#332

—

126

C₂×#332

—

127

C₆₇

#9700

yes

characteristic, complemented, Sylow

128

C₂×C₄⋊Q₈

C₂×D₆₇

yes

characteristic

129

C₂×C₄⋊Q₈

C₂×D₆₇

yes

characteristic

130

C₂×C₄⋊C₈

—

131

C₂×C₄⋊C₈

—

132

C₂×C₄×C₈

—

134

—

133

#391

—

134

#391

—

135

#391

—

136

#382

Dic₆₇

yes

—

137

C₄³

C₂×D₆₇

yes

characteristic

138

#336

—

134

—

139

#336

—

134

—

140

#332

—

134

—

141

#332

—

134

—

142

C₂²×Q₈

—

143

C₂²×Q₈

—

144

C₂²×C₈

—

134

—

145

C₂²×C₈

—

134

—

146

C₄⋊Q₈

—

147

C₄⋊Q₈

—

148

C₄⋊Q₈

—

149

C₄⋊Q₈

—

150

C₂×C₄⋊C₄

—

151

C₂×C₄⋊C₄

—

152

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

153

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

154

C₂×C₄²

C₂×Dic₆₇

yes

—

155

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

156

C₂×C₄²

C₂²⋊D₆₇

yes

—

157

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

158

C₂×C₄²

C₂²×D₆₇

yes

characteristic

159

C₄⋊C₈

—

160

C₄⋊C₈

—

134

—

161

C₄⋊C₈

—

134

—

162

C₄⋊C₈

—

134

—

163

C₄×C₈

—

268

—

164

#95

—

165

#95

—

166

C₂×Q₈

—

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂×Q₈

—

172

C₂×Q₈

—

173

C₂²×C₄

—

174

C₂²×C₄

—

175

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

176

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

177

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

178

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

179

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

180

C₂×C₈

—

268

—

181

C₂×C₈

—

268

—

182

C₂×C₈

—

268

—

183

C₂×C₈

—

268

—

184

C₂×C₈

—

134

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₄²

—

191

C₄²

—

192

C₄²

—

193

C₄²

—

194

C₄²

—

195

C₄²

D₄×D₆₇

yes

—

196

C₄²

#11761627

yes

—

197

C₂³

C₂×C₂²⋊Dic₆₇

yes

characteristic

198

Q₈

—

199

Q₈

—

200

Q₈

—

201

Q₈

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

#14523962

yes

—

214

C₂×C₄

D₄×Dic₆₇

yes

—

215

C₂×C₄

#14524014

yes

—

216

C₂×C₄

#14524030

yes

characteristic

217

C₂×C₄

#14524039

yes

characteristic

218

C₂×C₄

#14524063

yes

—

219

C₂×C₄

#14524063

yes

—

220

C₂×C₄

#14524068

yes

characteristic

221

C₂×C₄

#14524073

yes

characteristic

222

C₈

—

268

—

223

C₈

—

536

—

224

C₂²

#16863299

yes

characteristic

225

C₂²

C₂×#14523962

yes

—

226

C₂²

#16863392

yes

characteristic

227

C₂²

#16863392

yes

—

228

C₂²

#16863397

yes

characteristic

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

—

233

C₄

—

234

C₄

#16863608

yes

—

235

C₄

#16863615

yes

—

236

C₄

#16863750

yes

—

237

C₄

#16863778

yes

—

238

C₂

#22558874

yes

—

239

C₂

#22558909

yes

—

240

C₂

#22561021

yes

characteristic

241

C₂

#22561054

yes

characteristic

242

C₂

#22561142

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2862	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2862	D₆₇
hypocenter	C₆₇	#9700
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₆₇

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2862

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

N	H ≅ G/N	type
127. C₆₇	#9700	semidirect

H ≅ G/N

type

127. C₆₇

#9700

semidirect

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

subgroup	index
#22560326	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₂×#16863142	2
C₆₇×#2862	2
#9700	67

subgroup

index

#22560326

C₄×C₄⋊(C₆₇⋊C₈)

C₂×#16863142

C₆₇×#2862

#9700

Subgroups — #30754904

Conjugacy classes of subgroups

Complements

Maximal subgroups