Group #30775188 — subgroups

Subgroups — #30775188

Order 17152.

462 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 121 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×#16863776	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863201	C₂	1	2	yes	maximal
3	C₂×#16863201	C₂	1	2	yes	maximal
4	#22560942	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	—	2	1	no	—
7	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
8	C₂×#14524043	—	2	1	no	—
9	#16863776	—	2	4	no	—
10	C₂×#14523977	C₂²	1	1	yes	characteristic
11	C₂×#14523977	C₂²	1	1	yes	characteristic
12	#16863294	—	2	1	no	—
13	#16863201	—	2	4	no	—
14	#16863201	—	2	4	no	—
15	#16863159	—	2	1	no	—
16	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₃₄	—	2	2	no	—
19	C₂×Dic₂₆₈	—	2	4	no	—
20	C₂×Dic₂₆₈	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	2	no	—
22	C₂×Dic₂₆₈	—	2	2	no	—
23	C₂×Dic₂₆₈	—	2	2	no	—
24	C₂×Dic₂₆₈	—	2	2	no	—
25	C₂×Dic₂₆₈	—	4	2	no	—
26	C₂×Dic₂₆₈	—	4	2	no	—
27	C₂×Dic₂₆₈	D₄	1	4	yes	—
28	#14524043	—	2	4	no	—
29	#14524043	—	4	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	1	no	—
31	C₂×C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×C₄⋊Dic₆₇	—	4	1	no	—
33	C₂×C₄⋊Dic₆₇	—	4	1	no	—
34	C₂×Dic₆₇⋊C₄	—	4	1	no	—
35	#14523977	—	2	2	no	—
36	#14523977	—	2	1	no	—
37	#14523977	D₄	1	2	yes	—
38	C₆₇⋊#95	—	4	1	no	—
39	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₆₇	—	2	2	no	—
43	C₂×C₁₆×C₆₇	—	2	2	no	—
44	C₂×C₁₆×C₆₇	—	2	2	no	—
45	C₄×C₈×C₆₇	—	2	2	no	—
46	C₂×Dic₁₃₄	—	2	4	no	—
47	C₂×Dic₁₃₄	—	4	2	no	—
48	C₂×Dic₁₃₄	—	4	2	no	—
49	C₂×Dic₁₃₄	—	4	2	no	—
50	C₂×Dic₁₃₄	—	4	2	no	—
51	C₂×Dic₁₃₄	—	4	2	no	—
52	C₂²×Dic₆₇	—	4	1	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	C₂²×Dic₆₇	—	4	1	no	—
55	Dic₂₆₈	—	2	4	no	—
56	Dic₂₆₈	—	2	4	no	—
57	Dic₂₆₈	—	4	8	no	—
58	Dic₂₆₈	—	4	2	no	—
59	Dic₂₆₈	—	4	2	no	—
60	C₄⋊Dic₆₇	—	2	2	no	—
61	C₄⋊Dic₆₇	—	4	4	no	—
62	C₄⋊Dic₆₇	—	4	4	no	—
63	C₄⋊Dic₆₇	—	4	1	no	—
64	C₄⋊Dic₆₇	—	4	1	no	—
65	C₄⋊Dic₆₇	—	4	1	no	—
66	Dic₆₇⋊C₄	—	8	2	no	—
67	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	—	2	1	no	—
71	C₂×C₈×C₆₇	—	2	1	no	—
72	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
73	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₆₇	#41	1	2	yes	—
76	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
77	C₄²×C₆₇	—	2	1	no	—
78	C₄²×C₆₇	—	2	1	no	—
79	C₁₆×C₆₇	—	4	2	no	—
80	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
86	C₂×C₄×C₆₇	Q₃₂	1	2	yes	—
87	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₂×Dic₆₇	—	4	4	no	—
94	C₂×Dic₆₇	—	4	2	no	—
95	C₂×Dic₆₇	—	8	2	no	—
96	C₂×Dic₆₇	—	8	2	no	—
97	C₂×Dic₆₇	—	8	1	no	—
98	C₂×Dic₆₇	—	8	1	no	—
99	C₂×Dic₆₇	—	8	1	no	—
100	C₂×Dic₆₇	—	8	1	no	—
101	C₂×Dic₆₇	—	8	1	no	—
102	Dic₁₃₄	—	4	4	no	—
103	Dic₁₃₄	—	4	4	no	—
104	Dic₁₃₄	—	8	2	no	—
105	Dic₁₃₄	—	8	2	no	—
106	C₈×C₆₇	—	2	2	no	—
107	C₈×C₆₇	#121	1	2	yes	—
108	C₈×C₆₇	#123	1	2	yes	—
109	C₂²×C₆₇	#447	1	1	yes	characteristic
110	C₂²×C₆₇	#459	1	1	yes	characteristic
111	C₂²×C₆₇	#486	1	1	yes	characteristic
112	C₂²×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
113	C₂²×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
114	C₂²×C₆₇	#510	1	1	yes	characteristic
115	C₂²×C₆₇	#510	1	1	yes	characteristic
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	#394	1	1	yes	characteristic
119	C₄×C₆₇	#450	1	1	yes	characteristic
120	C₄×C₆₇	#460	1	1	yes	characteristic
121	C₄×C₆₇	#490	1	1	yes	characteristic
122	Dic₆₇	—	8	2	no	—
123	Dic₆₇	—	8	4	no	—
124	Dic₆₇	—	16	1	no	—
125	#13409	—	67	1	no	maximal, Sylow
126	C₂×C₆₇	#2041	1	1	yes	characteristic
127	C₂×C₆₇	#2185	1	1	yes	characteristic
128	C₂×C₆₇	#2188	1	1	yes	characteristic
129	C₂×C₆₇	#2208	1	1	yes	characteristic
130	C₂×C₆₇	#2209	1	1	yes	characteristic
131	C₂×C₆₇	#2241	1	1	yes	characteristic
132	C₂×C₆₇	#2245	1	1	yes	characteristic
133	C₂×#494	—	67	1	no	—
134	C₂×#358	—	67	2	no	—
135	C₂×#358	—	67	2	no	—
136	#1942	—	67	1	no	—
137	#1374	D₆₇	1	1	yes	characteristic
138	C₆₇	#13409	1	1	yes	characteristic, complemented, Sylow
139	C₂²×Q₁₆	—	67	2	no	—
140	C₂²×Q₁₆	—	134	1	no	—
141	C₂×C₄⋊Q₈	—	134	1	no	—
142	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
143	#494	—	134	4	no	—
144	C₂×#107	—	67	1	no	—
145	C₂×#107	—	67	1	no	—
146	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
147	#389	—	134	1	no	—
148	#358	—	134	4	no	—
149	#358	—	134	4	no	—
150	#340	—	134	1	no	—
151	C₂²×Q₈	—	134	2	no	—
152	C₂×Q₁₆	—	268	2	no	—
153	C₂×Q₁₆	—	268	2	no	—
154	C₂×Q₁₆	—	67	4	no	—
155	C₂×Q₁₆	—	134	4	no	—
156	C₂×Q₁₆	—	134	2	no	—
157	C₂×Q₁₆	—	134	2	no	—
158	C₂×Q₁₆	—	134	2	no	—
159	C₂×Q₁₆	—	134	2	no	—
160	C₂×Q₁₆	—	134	2	no	—
161	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
163	C₄⋊Q₈	—	268	2	no	—
164	C₄⋊Q₈	—	134	4	no	—
165	C₂×C₄⋊C₄	—	268	1	no	—
166	C₂×C₄⋊C₄	—	268	1	no	—
167	C₂×C₄⋊C₄	—	268	1	no	—
168	C₂×C₄⋊C₄	—	134	1	no	—
169	C₂×C₄⋊C₄	—	134	1	no	—
170	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	#107	—	67	2	no	—
175	#107	—	134	2	no	—
176	#107	—	134	1	no	—
177	C₄×C₈	—	2	2	no	—
178	#95	—	268	1	no	—
179	C₂×Q₈	—	268	2	no	—
180	C₂×Q₈	—	268	2	no	—
181	C₂×Q₈	—	268	2	no	—
182	C₂×Q₈	—	268	2	no	—
183	C₂×Q₈	—	268	2	no	—
184	C₂×Q₈	—	134	4	no	—
185	C₂²×C₄	—	268	1	no	—
186	C₂²×C₄	—	268	2	no	—
187	C₂²×C₄	—	268	1	no	—
188	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
189	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
191	Q₁₆	—	268	8	no	—
192	Q₁₆	—	268	2	no	—
193	Q₁₆	—	268	2	no	—
194	Q₁₆	—	134	4	no	—
195	Q₁₆	—	134	4	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	#11761617	1	2	yes	—
199	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
200	C₂×C₈	D₄×D₆₇	1	2	yes	—
201	C₂×C₈	#11761628	1	2	yes	—
202	C₂×C₈	#11761629	1	1	yes	characteristic
203	C₄⋊C₄	—	268	4	no	—
204	C₄⋊C₄	—	268	4	no	—
205	C₄⋊C₄	—	268	1	no	—
206	C₄⋊C₄	—	268	1	no	—
207	C₄⋊C₄	—	268	1	no	—
208	C₄⋊C₄	—	536	2	no	—
209	C₄⋊C₄	—	134	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₁₆	—	4	2	no	—
213	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
214	Q₈	—	268	4	no	—
215	Q₈	—	268	4	no	—
216	Q₈	—	536	2	no	—
217	Q₈	—	536	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	268	4	no	—
223	C₂×C₄	—	268	2	no	—
224	C₂×C₄	—	536	1	no	—
225	C₂×C₄	—	536	1	no	—
226	C₂×C₄	—	536	2	no	—
227	C₂×C₄	—	536	2	no	—
228	C₂×C₄	—	536	1	no	—
229	C₂×C₄	—	536	1	no	—
230	C₂×C₄	—	536	1	no	—
231	C₂×C₄	#14523988	1	2	yes	—
232	C₂×C₄	Dic₅₃₆	1	2	yes	—
233	C₂×C₄	#14524013	1	1	yes	characteristic
234	C₂×C₄	#14524018	1	1	yes	characteristic
235	C₂×C₄	#14524031	1	1	yes	characteristic
236	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
237	C₂×C₄	#14524071	1	1	yes	characteristic
238	C₂×C₄	#14524071	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	#14524016	1	2	yes	—
241	C₈	#14524028	1	2	yes	—
242	C₂²	#16863532	1	1	yes	characteristic
243	C₂²	#16863592	1	1	yes	characteristic
244	C₂²	#16863743	1	1	yes	characteristic
245	C₂²	C₂×#14523988	1	1	yes	characteristic
246	C₂²	C₂×Dic₅₃₆	1	1	yes	characteristic
247	C₂²	#16863839	1	1	yes	characteristic
248	C₂²	#16863839	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	536	4	no	—
252	C₄	—	536	2	no	—
253	C₄	—	1072	1	no	—
254	C₄	#16863314	1	1	yes	characteristic
255	C₄	#16863549	1	1	yes	characteristic
256	C₄	#16863597	1	1	yes	characteristic
257	C₄	#16863761	1	1	yes	characteristic
258	C₂	#22561444	1	1	yes	characteristic
259	C₂	#22562128	1	1	yes	characteristic
260	C₂	#22562145	1	1	yes	characteristic
261	C₂	#22562255	1	1	yes	characteristic
262	C₂	#22562260	1	1	yes	characteristic
263	C₂	#22562437	1	1	yes	characteristic
264	C₂	#22562455	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863776

C₂

yes

maximal, characteristic

C₂×#16863201

C₂

yes

maximal

C₂×#16863201

C₂

yes

maximal

#22560942

C₂

yes

maximal, characteristic

C₆₇×#1374

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₈

—

C₂²×Dic₂₆₈

C₂²

yes

—

C₂×#14524043

—

#16863776

—

C₂×#14523977

C₂²

yes

characteristic

C₂×#14523977

C₂²

yes

—

—

—

—

C₂²×C₁₆×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂²×Dic₁₃₄

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

D₄

yes

—

#14524043

—

#14524043

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×Dic₆₇⋊C₄

—

#14523977

—

#14523977

—

#14523977

D₄

yes

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₂²×C₄×C₆₇

D₈

yes

characteristic

C₂²×C₄×C₆₇

D₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

SD₃₂

yes

—

C₂×C₄×C₆₇

Q₃₂

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×SD₁₆

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

100

C₂×Dic₆₇

—

101

C₂×Dic₆₇

—

102

Dic₁₃₄

—

103

Dic₁₃₄

—

104

Dic₁₃₄

—

105

Dic₁₃₄

—

106

C₈×C₆₇

—

107

C₈×C₆₇

#121

yes

—

108

C₈×C₆₇

#123

yes

—

109

C₂²×C₆₇

#447

yes

characteristic

110

C₂²×C₆₇

#459

yes

characteristic

111

C₂²×C₆₇

#486

yes

characteristic

112

C₂²×C₆₇

C₂×SD₃₂

yes

characteristic

113

C₂²×C₆₇

C₂×Q₃₂

yes

characteristic

114

C₂²×C₆₇

#510

yes

characteristic

115

C₂²×C₆₇

#510

yes

characteristic

116

C₄×C₆₇

—

117

C₄×C₆₇

—

118

C₄×C₆₇

#394

yes

characteristic

119

C₄×C₆₇

#450

yes

characteristic

120

C₄×C₆₇

#460

yes

characteristic

121

C₄×C₆₇

#490

yes

characteristic

122

Dic₆₇

—

123

Dic₆₇

—

124

Dic₆₇

—

125

#13409

—

maximal, Sylow

126

C₂×C₆₇

#2041

yes

characteristic

127

C₂×C₆₇

#2185

yes

characteristic

128

C₂×C₆₇

#2188

yes

characteristic

129

C₂×C₆₇

#2208

yes

characteristic

130

C₂×C₆₇

#2209

yes

characteristic

131

C₂×C₆₇

#2241

yes

characteristic

132

C₂×C₆₇

#2245

yes

characteristic

133

C₂×#494

—

134

C₂×#358

—

135

C₂×#358

—

136

#1942

—

137

#1374

D₆₇

yes

characteristic

138

C₆₇

#13409

yes

characteristic, complemented, Sylow

139

C₂²×Q₁₆

—

140

C₂²×Q₁₆

—

134

—

141

C₂×C₄⋊Q₈

—

134

—

142

C₂²×C₁₆

C₂×D₆₇

yes

—

143

#494

—

134

—

144

C₂×#107

—

145

C₂×#107

—

146

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

147

#389

—

134

—

148

#358

—

134

—

149

#358

—

134

—

150

#340

—

134

—

151

C₂²×Q₈

—

134

—

152

C₂×Q₁₆

—

268

—

153

C₂×Q₁₆

—

268

—

154

C₂×Q₁₆

—

155

C₂×Q₁₆

—

134

—

156

C₂×Q₁₆

—

134

—

157

C₂×Q₁₆

—

134

—

158

C₂×Q₁₆

—

134

—

159

C₂×Q₁₆

—

134

—

160

C₂×Q₁₆

—

134

—

161

C₂²×C₈

D₂₆₈

yes

characteristic

162

C₂²×C₈

C₂²×D₆₇

yes

characteristic

163

C₄⋊Q₈

—

268

—

164

C₄⋊Q₈

—

134

—

165

C₂×C₄⋊C₄

—

268

—

166

C₂×C₄⋊C₄

—

268

—

167

C₂×C₄⋊C₄

—

268

—

168

C₂×C₄⋊C₄

—

134

—

169

C₂×C₄⋊C₄

—

134

—

170

C₂×C₄²

D₂₆₈

yes

characteristic

171

C₂×C₁₆

—

172

C₂×C₁₆

—

173

C₂×C₁₆

—

174

#107

—

175

#107

—

134

—

176

#107

—

134

—

177

C₄×C₈

—

178

#95

—

268

—

179

C₂×Q₈

—

268

—

180

C₂×Q₈

—

268

—

181

C₂×Q₈

—

268

—

182

C₂×Q₈

—

268

—

183

C₂×Q₈

—

268

—

184

C₂×Q₈

—

134

—

185

C₂²×C₄

—

268

—

186

C₂²×C₄

—

268

—

187

C₂²×C₄

—

268

—

188

C₂²×C₄

D₅₃₆

yes

characteristic

189

C₂²×C₄

D₅₃₆

yes

characteristic

190

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

191

Q₁₆

—

268

—

192

Q₁₆

—

268

—

193

Q₁₆

—

268

—

194

Q₁₆

—

134

—

195

Q₁₆

—

134

—

196

C₂×C₈

—

197

C₂×C₈

—

198

C₂×C₈

#11761617

yes

—

199

C₂×C₈

D₄×D₆₇

yes

characteristic

200

C₂×C₈

D₄×D₆₇

yes

—

201

C₂×C₈

#11761628

yes

—

202

C₂×C₈

#11761629

yes

characteristic

203

C₄⋊C₄

—

268

—

204

C₄⋊C₄

—

268

—

205

C₄⋊C₄

—

268

—

206

C₄⋊C₄

—

268

—

207

C₄⋊C₄

—

268

—

208

C₄⋊C₄

—

536

—

209

C₄⋊C₄

—

134

—

210

C₄²

—

211

C₄²

—

212

C₁₆

—

213

C₂³

C₂×D₅₃₆

yes

characteristic

214

Q₈

—

268

—

215

Q₈

—

268

—

216

Q₈

—

536

—

217

Q₈

—

536

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

—

222

C₂×C₄

—

268

—

223

C₂×C₄

—

268

—

224

C₂×C₄

—

536

—

225

C₂×C₄

—

536

—

226

C₂×C₄

—

536

—

227

C₂×C₄

—

536

—

228

C₂×C₄

—

536

—

229

C₂×C₄

—

536

—

230

C₂×C₄

—

536

—

231

C₂×C₄

#14523988

yes

—

232

C₂×C₄

Dic₅₃₆

yes

—

233

C₂×C₄

#14524013

yes

characteristic

234

C₂×C₄

#14524018

yes

characteristic

235

C₂×C₄

#14524031

yes

characteristic

236

C₂×C₄

C₂×#11761617

yes

characteristic

237

C₂×C₄

#14524071

yes

characteristic

238

C₂×C₄

#14524071

yes

characteristic

239

C₈

—

240

C₈

#14524016

yes

—

241

C₈

#14524028

yes

—

242

C₂²

#16863532

yes

characteristic

243

C₂²

#16863592

yes

characteristic

244

C₂²

#16863743

yes

characteristic

245

C₂²

C₂×#14523988

yes

characteristic

246

C₂²

C₂×Dic₅₃₆

yes

characteristic

247

C₂²

#16863839

yes

characteristic

248

C₂²

#16863839

yes

characteristic

249

C₄

—

250

C₄

—

251

C₄

—

536

—

252

C₄

—

536

—

253

C₄

—

1072

—

254

C₄

#16863314

yes

characteristic

255

C₄

#16863549

yes

characteristic

256

C₄

#16863597

yes

characteristic

257

C₄

#16863761

yes

characteristic

258

C₂

#22561444

yes

characteristic

259

C₂

#22562128

yes

characteristic

260

C₂

#22562145

yes

characteristic

261

C₂

#22562255

yes

characteristic

262

C₂

#22562260

yes

characteristic

263

C₂

#22562437

yes

characteristic

264

C₂

#22562455

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1374	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	#1374	D₆₇
hypocenter	C₆₇	#13409
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

subgroup

G/H

center

C₂³

C₂×D₅₃₆

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1374

C₂

socle, prime radical

C₂³×C₆₇

C₂×D₈

D₆₇

C₆₇

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

N	H ≅ G/N	type
138. C₆₇	#13409	semidirect

H ≅ G/N

type

138. C₆₇

#13409

semidirect

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

subgroup	index
C₂×#16863776	2
C₂×#16863201	2
#22560942	2
C₆₇×#1374	2
#13409	67

subgroup

index

C₆₇×#1374

Subgroups — #30775188

Conjugacy classes of subgroups

Complements

Maximal subgroups