Group #30748967 — subgroups

Subgroups — #30748967

Order 17152.

378 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 123 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×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22558957	C₂	1	1	yes	maximal, characteristic
3	#22558637	C₂	1	2	yes	maximal
4	#22558104	C₂	1	2	yes	maximal
5	C₆₇×#1837	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₃₄	—	2	1	no	—
8	C₈×Dic₁₃₄	—	2	1	no	—
9	C₈×Dic₁₃₄	C₄	1	2	yes	—
10	C₈×Dic₁₃₄	C₄	1	2	yes	—
11	#16863449	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₆₇⋊C₈	—	2	1	no	—
14	C₂×C₈×Dic₆₇	—	2	1	no	—
15	#16863149	—	2	1	no	—
16	#16863147	—	2	1	no	—
17	#16863126	—	2	4	no	—
18	#423×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#341×C₆₇	—	2	1	no	—
21	#341×C₆₇	—	2	1	no	—
22	C₂²×Dic₁₃₄	Q₈	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	—	2	1	no	—
24	C₄×Dic₁₃₄	—	2	1	no	—
25	C₄×Dic₁₃₄	—	2	1	no	—
26	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
27	C₄×Dic₁₃₄	D₄	1	2	yes	—
28	C₂×C₄⋊Dic₆₇	Q₈	1	1	yes	characteristic
29	C₂×Dic₆₇⋊C₄	—	2	1	no	—
30	C₂×C₄×Dic₆₇	—	2	1	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₆₇⋊C₈	—	2	2	no	—
35	Dic₆₇⋊C₈	—	2	2	no	—
36	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
37	C₄×C₆₇⋊C₈	—	2	2	no	—
38	C₈×Dic₆₇	—	2	2	no	—
39	C₈×Dic₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₆₇	—	2	2	no	—
45	C₄⋊C₈×C₆₇	—	2	2	no	—
46	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂×Dic₁₃₄	—	2	1	no	—
51	C₂×Dic₁₃₄	—	2	4	no	—
52	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
53	C₂²×Dic₆₇	—	2	1	no	—
54	C₂²×Dic₆₇	—	2	1	no	—
55	C₂×C₆₇⋊C₈	—	2	2	no	—
56	C₂×C₆₇⋊C₈	—	4	2	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₄⋊Dic₆₇	—	2	1	no	—
59	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
60	Dic₆₇⋊C₄	—	2	2	no	—
61	Dic₆₇⋊C₄	—	2	2	no	—
62	C₄×Dic₆₇	—	2	2	no	—
63	C₄×Dic₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
67	M₁₆×C₆₇	—	4	2	no	—
68	M₁₆×C₆₇	—	4	2	no	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	4	2	no	—
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₆₇	#41	1	1	yes	characteristic
80	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	4	1	no	—
83	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
86	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#135	1	2	yes	—
90	C₂×C₄×C₆₇	#137	1	2	yes	—
91	C₂×Dic₆₇	—	2	4	no	—
92	C₂×Dic₆₇	—	2	2	no	—
93	C₂×Dic₆₇	—	2	2	no	—
94	C₂×Dic₆₇	—	4	2	no	—
95	Dic₁₃₄	—	2	4	no	—
96	Dic₁₃₄	—	2	2	no	—
97	Dic₁₃₄	#108	1	4	yes	—
98	C₈×C₆₇	—	2	1	no	—
99	C₈×C₆₇	—	2	1	no	—
100	C₈×C₆₇	—	2	1	no	—
101	C₈×C₆₇	—	2	1	no	—
102	C₈×C₆₇	—	8	2	no	—
103	C₆₇⋊C₈	—	4	2	no	—
104	C₆₇⋊C₈	—	8	2	no	—
105	C₂²×C₆₇	—	2	1	no	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	#380	1	1	yes	characteristic
108	C₂²×C₆₇	#479	1	2	yes	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	#416	1	1	yes	characteristic
113	C₄×C₆₇	#419	1	1	yes	characteristic
114	C₄×C₆₇	C₂×#108	1	2	yes	—
115	C₄×C₆₇	#464	1	2	yes	—
116	Dic₆₇	—	2	4	no	—
117	Dic₆₇	—	4	2	no	—
118	#8661	—	67	1	no	maximal, Sylow
119	C₂×C₆₇	—	2	2	no	—
120	C₂×C₆₇	#1805	1	1	yes	characteristic
121	C₂×C₆₇	#1810	1	1	yes	characteristic
122	C₂×C₆₇	#1814	1	1	yes	characteristic
123	C₂×C₈×Q₈	—	67	1	no	—
124	#1837	D₆₇	1	1	yes	characteristic
125	#1564	—	67	1	no	—
126	#1487	—	67	2	no	—
127	#1301	—	67	2	no	—
128	C₆₇	#8661	1	1	yes	characteristic, complemented, Sylow
129	C₂×C₄×Q₈	—	67	1	no	—
130	C₈×Q₈	—	67	2	no	—
131	C₈×Q₈	—	67	2	no	—
132	C₈×Q₈	—	134	1	no	—
133	C₈×Q₈	—	134	1	no	—
134	#431	—	67	2	no	—
135	#423	C₂×D₆₇	1	2	yes	—
136	C₂×C₄⋊C₈	—	67	1	no	—
137	C₂×C₄⋊C₈	—	134	1	no	—
138	C₂×C₄×C₈	—	134	1	no	—
139	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
140	#341	—	2	1	no	—
141	#341	—	2	1	no	—
142	#335	—	134	1	no	—
143	#334	—	134	1	no	—
144	Q₈⋊C₈	—	134	4	no	—
145	C₂²×Q₈	—	67	1	no	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂²×C₈	—	134	1	no	—
148	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
149	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
150	C₄×Q₈	—	67	2	no	—
151	C₄×Q₈	—	67	2	no	—
152	C₄×Q₈	—	134	1	no	—
153	C₄×Q₈	—	134	1	no	—
154	C₂×C₄⋊C₄	—	67	1	no	—
155	C₂×C₄⋊C₄	—	134	1	no	—
156	C₂×C₄²	—	134	1	no	—
157	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	67	2	no	—
161	C₄⋊C₈	—	134	2	no	—
162	C₄⋊C₈	—	134	2	no	—
163	C₄⋊C₈	—	134	2	no	—
164	C₄⋊C₈	—	134	1	no	—
165	C₂²⋊C₈	—	134	2	no	—
166	C₄×C₈	—	134	2	no	—
167	C₄×C₈	—	134	2	no	—
168	C₄×C₈	—	134	2	no	—
169	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
170	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
171	C₄×C₈	D₂₆₈	1	1	yes	characteristic
172	C₄×C₈	D₂₆₈	1	1	yes	characteristic
173	C₂×Q₈	—	67	2	no	—
174	C₂×Q₈	—	134	4	no	—
175	C₂×Q₈	—	134	1	no	—
176	C₂²×C₄	—	134	1	no	—
177	C₂²×C₄	—	134	1	no	—
178	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
179	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
180	C₂²×C₄	#11761629	1	1	yes	characteristic
181	M₁₆	—	4	2	no	—
182	M₁₆	—	4	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	268	2	no	—
187	C₂×C₈	—	268	2	no	—
188	C₂×C₈	—	134	2	no	—
189	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
190	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
191	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
192	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
193	C₄⋊C₄	—	67	2	no	—
194	C₄⋊C₄	—	134	2	no	—
195	C₄⋊C₄	—	134	2	no	—
196	C₄⋊C₄	—	134	1	no	—
197	C₄²	—	134	2	no	—
198	C₄²	—	134	2	no	—
199	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
200	C₄²	D₄×D₆₇	1	1	yes	characteristic
201	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
202	C₄²	#11761627	1	1	yes	characteristic
203	C₂³	#14524019	1	1	yes	characteristic
204	Q₈	—	67	4	no	—
205	Q₈	—	134	4	no	—
206	Q₈	—	134	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	268	2	no	—
210	C₂×C₄	—	134	2	no	—
211	C₂×C₄	—	134	4	no	—
212	C₂×C₄	—	134	2	no	—
213	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
214	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
215	C₂×C₄	#14524001	1	1	yes	characteristic
216	C₂×C₄	#14524014	1	1	yes	characteristic
217	C₂×C₄	#14524018	1	1	yes	characteristic
218	C₂×C₄	#14524023	1	1	yes	characteristic
219	C₂×C₄	#14524063	1	2	yes	—
220	C₂×C₄	#14524073	1	2	yes	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	8	2	no	—
226	C₈	—	268	2	no	—
227	C₈	—	536	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	#16863253	1	1	yes	characteristic
231	C₂²	#16863700	1	2	yes	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	268	2	no	—
236	C₄	—	134	4	no	—
237	C₄	#16863390	1	1	yes	characteristic
238	C₄	#16863404	1	1	yes	characteristic
239	C₄	#108×D₆₇	1	2	yes	—
240	C₄	#16863615	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#22560221	1	1	yes	characteristic
243	C₂	#22560244	1	1	yes	characteristic
244	C₂	#22560264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₃₄

C₂

yes

maximal, characteristic

#22558957

C₂

yes

maximal, characteristic

#22558637

C₂

yes

maximal

#22558104

C₂

yes

maximal

C₆₇×#1837

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₃₄

C₂²

yes

characteristic

C₈×Dic₁₃₄

—

C₈×Dic₁₃₄

—

C₈×Dic₁₃₄

C₄

yes

—

C₈×Dic₁₃₄

C₄

yes

—

#16863449

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₂×Dic₆₇⋊C₈

—

C₂×C₈×Dic₆₇

—

#16863149

—

#16863147

—

#16863126

—

#423×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#341×C₆₇

—

#341×C₆₇

—

C₂²×Dic₁₃₄

Q₈

yes

characteristic

C₂²×C₆₇⋊C₈

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

C₂×C₄

yes

—

C₄×Dic₁₃₄

D₄

yes

—

C₂×C₄⋊Dic₆₇

Q₈

yes

characteristic

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

C₂×C₄

yes

—

Dic₆₇⋊C₈

—

Dic₆₇⋊C₈

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₈×Dic₆₇

—

C₈×Dic₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₄⋊C₄

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₄⋊C₄

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

characteristic

C₂³×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

#108

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

100

C₈×C₆₇

—

101

C₈×C₆₇

—

102

C₈×C₆₇

—

103

C₆₇⋊C₈

—

104

C₆₇⋊C₈

—

105

C₂²×C₆₇

—

106

C₂²×C₆₇

—

107

C₂²×C₆₇

#380

yes

characteristic

108

C₂²×C₆₇

#479

yes

—

109

C₄×C₆₇

—

110

C₄×C₆₇

—

111

C₄×C₆₇

—

112

C₄×C₆₇

#416

yes

characteristic

113

C₄×C₆₇

#419

yes

characteristic

114

C₄×C₆₇

C₂×#108

yes

—

115

C₄×C₆₇

#464

yes

—

116

Dic₆₇

—

117

Dic₆₇

—

118

#8661

—

maximal, Sylow

119

C₂×C₆₇

—

120

C₂×C₆₇

#1805

yes

characteristic

121

C₂×C₆₇

#1810

yes

characteristic

122

C₂×C₆₇

#1814

yes

characteristic

123

C₂×C₈×Q₈

—

124

#1837

D₆₇

yes

characteristic

125

#1564

—

126

#1487

—

127

#1301

—

128

C₆₇

#8661

yes

characteristic, complemented, Sylow

129

C₂×C₄×Q₈

—

130

C₈×Q₈

—

131

C₈×Q₈

—

132

C₈×Q₈

—

134

—

133

C₈×Q₈

—

134

—

134

#431

—

135

#423

C₂×D₆₇

yes

—

136

C₂×C₄⋊C₈

—

137

C₂×C₄⋊C₈

—

134

—

138

C₂×C₄×C₈

—

134

—

139

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

140

#341

—

141

#341

—

142

#335

—

134

—

143

#334

—

134

—

144

Q₈⋊C₈

—

134

—

145

C₂²×Q₈

—

146

C₂×M₁₆

—

147

C₂²×C₈

—

134

—

148

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

149

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

150

C₄×Q₈

—

151

C₄×Q₈

—

152

C₄×Q₈

—

134

—

153

C₄×Q₈

—

134

—

154

C₂×C₄⋊C₄

—

155

C₂×C₄⋊C₄

—

134

—

156

C₂×C₄²

—

134

—

157

C₂×C₄²

C₂²×D₆₇

yes

characteristic

158

C₄⋊C₈

—

159

C₄⋊C₈

—

160

C₄⋊C₈

—

161

C₄⋊C₈

—

134

—

162

C₄⋊C₈

—

134

—

163

C₄⋊C₈

—

134

—

164

C₄⋊C₈

—

134

—

165

C₂²⋊C₈

—

134

—

166

C₄×C₈

—

134

—

167

C₄×C₈

—

134

—

168

C₄×C₈

—

134

—

169

C₄×C₈

C₄×D₆₇

yes

characteristic

170

C₄×C₈

C₄×D₆₇

yes

characteristic

171

C₄×C₈

D₂₆₈

yes

characteristic

172

C₄×C₈

D₂₆₈

yes

characteristic

173

C₂×Q₈

—

174

C₂×Q₈

—

134

—

175

C₂×Q₈

—

134

—

176

C₂²×C₄

—

134

—

177

C₂²×C₄

—

134

—

178

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

179

C₂²×C₄

Q₈×D₆₇

yes

characteristic

180

C₂²×C₄

#11761629

yes

characteristic

181

M₁₆

—

182

M₁₆

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₂×C₈

—

186

C₂×C₈

—

268

—

187

C₂×C₈

—

268

—

188

C₂×C₈

—

134

—

189

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

190

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

191

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

192

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

193

C₄⋊C₄

—

194

C₄⋊C₄

—

134

—

195

C₄⋊C₄

—

134

—

196

C₄⋊C₄

—

134

—

197

C₄²

—

134

—

198

C₄²

—

134

—

199

C₄²

C₂×C₄×D₆₇

yes

characteristic

200

C₄²

D₄×D₆₇

yes

characteristic

201

C₄²

C₂×D₂₆₈

yes

characteristic

202

C₄²

#11761627

yes

characteristic

203

C₂³

#14524019

yes

characteristic

204

Q₈

—

205

Q₈

—

134

—

206

Q₈

—

134

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

268

—

210

C₂×C₄

—

134

—

211

C₂×C₄

—

134

—

212

C₂×C₄

—

134

—

213

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

214

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

215

C₂×C₄

#14524001

yes

characteristic

216

C₂×C₄

#14524014

yes

characteristic

217

C₂×C₄

#14524018

yes

characteristic

218

C₂×C₄

#14524023

yes

characteristic

219

C₂×C₄

#14524063

yes

—

220

C₂×C₄

#14524073

yes

—

221

C₈

—

222

C₈

—

223

C₈

—

224

C₈

—

225

C₈

—

226

C₈

—

268

—

227

C₈

—

536

—

228

C₂²

—

229

C₂²

—

230

C₂²

#16863253

yes

characteristic

231

C₂²

#16863700

yes

—

232

C₄

—

233

C₄

—

234

C₄

—

235

C₄

—

268

—

236

C₄

—

134

—

237

C₄

#16863390

yes

characteristic

238

C₄

#16863404

yes

characteristic

239

C₄

#108×D₆₇

yes

—

240

C₄

#16863615

yes

—

241

C₂

—

242

C₂

#22560221

yes

characteristic

243

C₂

#22560244

yes

characteristic

244

C₂

#22560264

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524014
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1837	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1837	D₆₇
hypocenter	C₆₇	#8661
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂×C₄

#14524014

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1837

C₂

socle

C₂²×C₆₇

#380

prime radical

C₂³×C₆₇

C₂²⋊Q₈

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
128. C₆₇	#8661	semidirect

H ≅ G/N

type

128. C₆₇

#8661

semidirect

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

subgroup	index
C₂×C₈×Dic₁₃₄	2
#22558957	2
#22558637	2
#22558104	2
C₆₇×#1837	2
#8661	67

subgroup

index

C₂×C₈×Dic₁₃₄

C₆₇×#1837

Subgroups — #30748967

Conjugacy classes of subgroups

Complements

Maximal subgroups