Group #30688686 — subgroups

Subgroups — C₆₇×#9435 #30688686

Order 17152.

504 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₆₇×#3152	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2979	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1560	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1549	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1299	C₂	1	2	yes	maximal, complemented
6	C₂×#137×C₆₇	—	2	2	no	—
7	#532×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
9	#497×C₆₇	C₄	1	4	yes	—
10	#494×C₆₇	—	2	1	no	—
11	#492×C₆₇	—	2	1	no	—
12	#436×C₆₇	—	2	2	no	—
13	C₈×D₄×C₆₇	—	2	2	no	—
14	#431×C₆₇	C₂²	1	2	yes	—
15	#423×C₆₇	C₂²	1	2	yes	—
16	#408×C₆₇	—	2	2	no	—
17	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	#333×C₆₇	—	2	1	no	—
19	#332×C₆₇	—	2	2	no	—
20	#332×C₆₇	—	2	2	no	—
21	#331×C₆₇	—	2	1	no	—
22	C₂×#41×C₆₇	D₄	1	2	yes	—
23	C₂²×Q₈×C₆₇	—	2	1	no	—
24	#137×C₆₇	—	4	4	no	—
25	#137×C₆₇	—	4	4	no	—
26	C₂×Q₁₆×C₆₇	—	2	4	no	—
27	C₂×SD₁₆×C₆₇	—	2	4	no	—
28	C₂×M₁₆×C₆₇	—	2	1	no	—
29	C₂×M₁₆×C₆₇	—	2	2	no	—
30	C₂×M₁₆×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	—	2	2	no	—
33	C₄⋊Q₈×C₆₇	—	2	1	no	—
34	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
36	#127×C₆₇	D₄	1	1	yes	characteristic
37	#124×C₆₇	C₂×C₄	1	2	yes	—
38	#121×C₆₇	—	2	2	no	—
39	C₄×D₄×C₆₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
45	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
46	C₂²⋊C₈×C₆₇	—	2	4	no	—
47	C₂²⋊C₈×C₆₇	—	2	2	no	—
48	#97×C₆₇	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₆₇	—	2	1	no	—
50	C₄×C₈×C₆₇	—	2	2	no	—
51	#41×C₆₇	—	4	2	no	—
52	#41×C₆₇	—	4	2	no	—
53	C₂×Q₈×C₆₇	—	2	2	no	—
54	C₂×Q₈×C₆₇	—	4	2	no	—
55	C₂×Q₈×C₆₇	—	4	2	no	—
56	C₂×Q₈×C₆₇	—	4	1	no	—
57	C₂×Q₈×C₆₇	C₂²⋊C₄	1	2	yes	—
58	C₂×D₄×C₆₇	—	2	2	no	—
59	C₂×D₄×C₆₇	—	2	2	no	—
60	C₂×D₄×C₆₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	4	1	no	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	#41	1	2	yes	—
65	Q₁₆×C₆₇	—	4	8	no	—
66	SD₁₆×C₆₇	—	4	8	no	—
67	M₁₆×C₆₇	—	2	2	no	—
68	M₁₆×C₆₇	—	2	2	no	—
69	M₁₆×C₆₇	—	8	2	no	—
70	M₁₆×C₆₇	—	8	2	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	—	8	2	no	—
76	C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₄⋊C₄×C₆₇	—	2	4	no	—
78	C₄⋊C₄×C₆₇	—	4	2	no	—
79	C₂²⋊C₄×C₆₇	—	2	4	no	—
80	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×D₄	1	2	yes	—
83	C₂³×C₆₇	—	2	2	no	—
84	C₂³×C₆₇	#123	1	1	yes	characteristic
85	Q₈×C₆₇	—	4	2	no	—
86	Q₈×C₆₇	—	4	4	no	—
87	Q₈×C₆₇	—	8	1	no	—
88	Q₈×C₆₇	—	8	1	no	—
89	D₄×C₆₇	—	4	2	no	—
90	D₄×C₆₇	—	4	2	no	—
91	D₄×C₆₇	—	4	2	no	—
92	D₄×C₆₇	—	8	2	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	4	2	no	—
99	C₂×C₄×C₆₇	—	4	2	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	—	8	2	no	—
102	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#121	1	2	yes	—
106	C₈×C₆₇	—	2	4	no	—
107	C₈×C₆₇	—	4	2	no	—
108	C₈×C₆₇	—	4	2	no	—
109	C₈×C₆₇	—	8	2	no	—
110	C₂²×C₆₇	—	2	1	no	—
111	C₂²×C₆₇	—	2	2	no	—
112	C₂²×C₆₇	—	4	2	no	—
113	C₂²×C₆₇	—	4	4	no	—
114	C₂²×C₆₇	#386	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	—	8	2	no	—
119	C₄×C₆₇	#453	1	1	yes	characteristic
120	C₄×C₆₇	#453	1	1	yes	characteristic
121	C₄×C₆₇	#471	1	2	yes	—
122	C₄×C₆₇	#473	1	2	yes	—
123	#9435	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₆₇	—	4	1	no	—
125	C₂×C₆₇	—	8	2	no	—
126	C₂×C₆₇	#1887	1	1	yes	characteristic
127	C₂×C₆₇	#1908	1	2	yes	—
128	#3152	C₂×C₆₇	1	1	yes	characteristic
129	#2979	C₂×C₆₇	1	1	yes	characteristic
130	#1560	C₂×C₆₇	1	1	yes	characteristic
131	#1549	C₂×C₆₇	1	2	yes	complemented
132	#1299	C₂×C₆₇	1	2	yes	complemented
133	C₆₇	#9435	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×#137	—	2	2	no	—
135	#532	C₂²×C₆₇	1	1	yes	characteristic
136	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
137	#497	C₄×C₆₇	1	4	yes	—
138	#494	—	2	1	no	—
139	#492	—	2	1	no	—
140	#436	—	2	2	no	—
141	C₈×D₄	—	2	2	no	—
142	#431	C₂²×C₆₇	1	2	yes	—
143	#423	C₂²×C₆₇	1	2	yes	—
144	#408	—	2	2	no	—
145	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
146	#333	—	2	1	no	—
147	#332	—	2	2	no	—
148	#332	—	2	2	no	—
149	#331	—	2	1	no	—
150	C₂×#41	D₄×C₆₇	1	2	yes	—
151	C₂²×Q₈	—	2	1	no	—
152	#137	—	4	4	no	—
153	#137	—	4	4	no	—
154	C₂×Q₁₆	—	2	4	no	—
155	C₂×SD₁₆	—	2	4	no	—
156	C₂×M₁₆	—	2	1	no	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	1	no	—
162	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
163	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
164	#127	D₄×C₆₇	1	1	yes	characteristic
165	#124	C₂×C₄×C₆₇	1	2	yes	—
166	#121	—	2	2	no	—
167	C₄×D₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
173	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
174	C₂²⋊C₈	—	2	4	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	#97	C₂×C₄×C₆₇	1	2	yes	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	—	2	2	no	—
179	#41	—	4	2	no	—
180	#41	—	4	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	1	no	—
185	C₂×Q₈	C₂²⋊C₄×C₆₇	1	2	yes	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	C₂²⋊C₄×C₆₇	1	2	yes	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	4	1	no	—
191	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
192	C₂²×C₄	#41×C₆₇	1	2	yes	—
193	Q₁₆	—	4	8	no	—
194	SD₁₆	—	4	8	no	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	2	no	—
197	M₁₆	—	8	2	no	—
198	M₁₆	—	8	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	8	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	2	4	no	—
208	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
209	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
210	C₄²	C₂×D₄×C₆₇	1	2	yes	—
211	C₂³	—	2	2	no	—
212	C₂³	#123×C₆₇	1	1	yes	characteristic
213	Q₈	—	4	2	no	—
214	Q₈	—	4	4	no	—
215	Q₈	—	8	1	no	—
216	Q₈	—	8	1	no	—
217	D₄	—	4	2	no	—
218	D₄	—	4	2	no	—
219	D₄	—	4	2	no	—
220	D₄	—	8	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
231	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
232	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
233	C₂×C₄	#121×C₆₇	1	2	yes	—
234	C₈	—	2	4	no	—
235	C₈	—	4	2	no	—
236	C₈	—	4	2	no	—
237	C₈	—	8	2	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	4	no	—
242	C₂²	#386×C₆₇	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	2	no	—
246	C₄	—	8	2	no	—
247	C₄	#453×C₆₇	1	1	yes	characteristic
248	C₄	#453×C₆₇	1	1	yes	characteristic
249	C₄	#471×C₆₇	1	2	yes	—
250	C₄	#473×C₆₇	1	2	yes	—
251	C₂	—	4	1	no	—
252	C₂	—	8	2	no	—
253	C₂	C₆₇×#1887	1	1	yes	characteristic
254	C₂	C₆₇×#1908	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#3152

C₂

yes

maximal, characteristic

C₆₇×#2979

C₂

yes

maximal, characteristic

C₆₇×#1560

C₂

yes

maximal, characteristic

C₆₇×#1549

C₂

yes

maximal, complemented

C₆₇×#1299

C₂

yes

maximal, complemented

C₂×#137×C₆₇

—

#532×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

#497×C₆₇

C₄

yes

—

#494×C₆₇

—

#492×C₆₇

—

#436×C₆₇

—

C₈×D₄×C₆₇

—

#431×C₆₇

C₂²

yes

—

#423×C₆₇

C₂²

yes

—

#408×C₆₇

—

C₄×M₁₆×C₆₇

C₂²

yes

characteristic

#333×C₆₇

—

#332×C₆₇

—

#332×C₆₇

—

#331×C₆₇

—

C₂×#41×C₆₇

D₄

yes

—

C₂²×Q₈×C₆₇

—

#137×C₆₇

—

#137×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×SD₁₆×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

C₄⋊Q₈×C₆₇

D₄

yes

characteristic

#127×C₆₇

D₄

yes

characteristic

#124×C₆₇

C₂×C₄

yes

—

#121×C₆₇

—

C₄×D₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

D₄

yes

—

C₄⋊C₈×C₆₇

D₄

yes

—

C₂²⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂²⋊C₄

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

Q₁₆×C₆₇

—

SD₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

#123

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

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

—

100

C₂×C₄×C₆₇

—

101

C₂×C₄×C₆₇

—

102

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

103

C₂×C₄×C₆₇

C₄×D₄

yes

—

104

C₂×C₄×C₆₇

#120

yes

characteristic

105

C₂×C₄×C₆₇

#121

yes

—

106

C₈×C₆₇

—

107

C₈×C₆₇

—

108

C₈×C₆₇

—

109

C₈×C₆₇

—

110

C₂²×C₆₇

—

111

C₂²×C₆₇

—

112

C₂²×C₆₇

—

113

C₂²×C₆₇

—

114

C₂²×C₆₇

#386

yes

characteristic

115

C₄×C₆₇

—

116

C₄×C₆₇

—

117

C₄×C₆₇

—

118

C₄×C₆₇

—

119

C₄×C₆₇

#453

yes

characteristic

120

C₄×C₆₇

#453

yes

characteristic

121

C₄×C₆₇

#471

yes

—

122

C₄×C₆₇

#473

yes

—

123

#9435

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

124

C₂×C₆₇

—

125

C₂×C₆₇

—

126

C₂×C₆₇

#1887

yes

characteristic

127

C₂×C₆₇

#1908

yes

—

128

#3152

C₂×C₆₇

yes

characteristic

129

#2979

C₂×C₆₇

yes

characteristic

130

#1560

C₂×C₆₇

yes

characteristic

131

#1549

C₂×C₆₇

yes

complemented

132

#1299

C₂×C₆₇

yes

complemented

133

C₆₇

#9435

yes

characteristic, direct product factor, Sylow

134

C₂×#137

—

135

#532

C₂²×C₆₇

yes

characteristic

136

C₂×C₄⋊Q₈

C₂²×C₆₇

yes

characteristic

137

#497

C₄×C₆₇

yes

—

138

#494

—

139

#492

—

140

#436

—

141

C₈×D₄

—

142

#431

C₂²×C₆₇

yes

—

143

#423

C₂²×C₆₇

yes

—

144

#408

—

145

C₄×M₁₆

C₂²×C₆₇

yes

characteristic

146

#333

—

147

#332

—

148

#332

—

149

#331

—

150

C₂×#41

D₄×C₆₇

yes

—

151

C₂²×Q₈

—

152

#137

—

153

#137

—

154

C₂×Q₁₆

—

155

C₂×SD₁₆

—

156

C₂×M₁₆

—

157

C₂×M₁₆

—

158

C₂×M₁₆

—

159

C₂²×C₈

—

160

C₄⋊Q₈

—

161

C₄⋊Q₈

—

162

C₄⋊Q₈

C₂×C₄×C₆₇

yes

—

163

C₄⋊Q₈

D₄×C₆₇

yes

characteristic

164

#127

D₄×C₆₇

yes

characteristic

165

#124

C₂×C₄×C₆₇

yes

—

166

#121

—

167

C₄×D₄

—

168

C₂×C₄⋊C₄

—

169

C₂×C₄²

C₂³×C₆₇

yes

characteristic

170

C₄⋊C₈

—

171

C₄⋊C₈

—

172

C₄⋊C₈

D₄×C₆₇

yes

—

173

C₄⋊C₈

D₄×C₆₇

yes

—

174

C₂²⋊C₈

—

175

C₂²⋊C₈

—

176

#97

C₂×C₄×C₆₇

yes

—

177

C₄×C₈

—

178

C₄×C₈

—

179

#41

—

180

#41

—

181

C₂×Q₈

—

182

C₂×Q₈

—

183

C₂×Q₈

—

184

C₂×Q₈

—

185

C₂×Q₈

C₂²⋊C₄×C₆₇

yes

—

186

C₂×D₄

—

187

C₂×D₄

—

188

C₂×D₄

C₂²⋊C₄×C₆₇

yes

—

189

C₂²×C₄

—

190

C₂²×C₄

—

191

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

192

C₂²×C₄

#41×C₆₇

yes

—

193

Q₁₆

—

194

SD₁₆

—

195

M₁₆

—

196

M₁₆

—

197

M₁₆

—

198

M₁₆

—

199

C₂×C₈

—

200

C₂×C₈

—

201

C₂×C₈

—

202

C₂×C₈

—

203

C₂×C₈

—

204

C₄⋊C₄

—

205

C₄⋊C₄

—

206

C₄⋊C₄

—

207

C₂²⋊C₄

—

208

C₄²

C₂²×C₄×C₆₇

yes

characteristic

209

C₄²

C₂×D₄×C₆₇

yes

characteristic

210

C₄²

C₂×D₄×C₆₇

yes

—

211

C₂³

—

212

C₂³

#123×C₆₇

yes

characteristic

213

Q₈

—

214

Q₈

—

215

Q₈

—

216

Q₈

—

217

D₄

—

218

D₄

—

219

D₄

—

220

D₄

—

221

C₂×C₄

—

222

C₂×C₄

—

223

C₂×C₄

—

224

C₂×C₄

—

225

C₂×C₄

—

226

C₂×C₄

—

227

C₂×C₄

—

228

C₂×C₄

—

229

C₂×C₄

—

230

C₂×C₄

C₂×C₂²⋊C₄×C₆₇

yes

characteristic

231

C₂×C₄

C₄×D₄×C₆₇

yes

—

232

C₂×C₄

#120×C₆₇

yes

characteristic

233

C₂×C₄

#121×C₆₇

yes

—

234

C₈

—

235

C₈

—

236

C₈

—

237

C₈

—

238

C₂²

—

239

C₂²

—

240

C₂²

—

241

C₂²

—

242

C₂²

#386×C₆₇

yes

characteristic

243

C₄

—

244

C₄

—

245

C₄

—

246

C₄

—

247

C₄

#453×C₆₇

yes

characteristic

248

C₄

#453×C₆₇

yes

characteristic

249

C₄

#471×C₆₇

yes

—

250

C₄

#473×C₆₇

yes

—

251

C₂

—

252

C₂

—

253

C₂

C₆₇×#1887

yes

characteristic

254

C₂

C₆₇×#1908

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₇	#386
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	#532×C₆₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₇

#386

commutator

C₄²

C₂²×C₄×C₆₇

Frattini, cosocle

C₂×C₄²

C₂³×C₆₇

prime radical

#532×C₆₇

C₂²

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
4. C₆₇×#1549	C₂	semidirect
5. C₆₇×#1299	C₂	semidirect
123. #9435	C₆₇	direct (N × H)
131. #1549	C₂×C₆₇	semidirect
132. #1299	C₂×C₆₇	semidirect
133. C₆₇	#9435	direct (N × H)

H ≅ G/N

type

4. C₆₇×#1549

C₂

semidirect

5. C₆₇×#1299

C₂

semidirect

123. #9435

C₆₇

direct (N × H)

131. #1549

C₂×C₆₇

semidirect

132. #1299

C₂×C₆₇

semidirect

133. C₆₇

#9435

direct (N × H)

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

subgroup	index
C₆₇×#3152	2
C₆₇×#2979	2
C₆₇×#1560	2
C₆₇×#1549	2
C₆₇×#1299	2
#9435	67

subgroup

index

C₆₇×#3152

C₆₇×#2979

C₆₇×#1560

C₆₇×#1549

C₆₇×#1299

#9435

Subgroups — C₆₇×#9435 #30688686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#9435 #30688686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9435 #30688686