Group #30712041 — subgroups

Subgroups — C₆₇×#32790 #30712041

Order 17152.

716 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 234 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×#2857	C₂	1	2	yes	maximal
2	C₆₇×#2100	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2076	C₂	1	4	yes	maximal
4	C₆₇×#2057	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1817	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
7	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
8	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#107×C₆₇	C₂²	1	4	yes	—
10	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
11	#391×C₆₇	—	2	2	no	—
12	#389×C₆₇	—	2	2	no	—
13	#389×C₆₇	C₂²	1	4	yes	—
14	#384×C₆₇	C₂²	1	2	yes	—
15	#380×C₆₇	—	2	2	no	—
16	#380×C₆₇	C₂²	1	4	yes	—
17	#340×C₆₇	C₂²	1	8	yes	—
18	#340×C₆₇	C₂²	1	4	yes	—
19	#336×C₆₇	C₂²	1	4	yes	—
20	C₂²×Q₈×C₆₇	—	2	2	no	—
21	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
25	C₄⋊Q₈×C₆₇	—	4	4	no	—
26	C₂²⋊Q₈×C₆₇	—	4	8	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
32	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
33	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
34	C₂×C₄²×C₆₇	—	4	2	no	—
35	C₂×C₄²×C₆₇	D₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
37	#107×C₆₇	—	4	4	no	—
38	C₂²⋊C₈×C₆₇	—	4	2	no	—
39	#95×C₆₇	—	2	4	no	—
40	#95×C₆₇	—	2	4	no	—
41	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂×Q₈×C₆₇	—	4	4	no	—
43	C₂×Q₈×C₆₇	—	4	4	no	—
44	C₂×Q₈×C₆₇	—	4	4	no	—
45	C₂²×C₄×C₆₇	—	2	4	no	—
46	C₂²×C₄×C₆₇	—	2	2	no	—
47	C₂²×C₄×C₆₇	—	2	4	no	—
48	C₂²×C₄×C₆₇	—	2	4	no	—
49	C₂²×C₄×C₆₇	—	2	4	no	—
50	C₂²×C₄×C₆₇	—	4	1	no	—
51	C₂²×C₄×C₆₇	—	4	1	no	—
52	C₂²×C₄×C₆₇	—	4	1	no	—
53	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	—	2	8	no	—
60	C₄⋊C₄×C₆₇	—	2	8	no	—
61	C₄⋊C₄×C₆₇	—	2	8	no	—
62	C₄⋊C₄×C₆₇	—	4	4	no	—
63	C₄⋊C₄×C₆₇	—	4	4	no	—
64	C₄⋊C₄×C₆₇	—	4	4	no	—
65	C₄⋊C₄×C₆₇	—	4	8	no	—
66	C₂²⋊C₄×C₆₇	—	4	8	no	—
67	C₄²×C₆₇	—	2	4	no	—
68	C₄²×C₆₇	—	4	2	no	—
69	C₂³×C₆₇	—	2	1	no	—
70	C₂³×C₆₇	—	2	2	no	—
71	C₂³×C₆₇	—	2	2	no	—
72	C₂³×C₆₇	—	2	1	no	—
73	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
74	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
75	Q₈×C₆₇	—	8	8	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	4	4	no	—
83	C₂×C₄×C₆₇	—	4	4	no	—
84	C₂×C₄×C₆₇	—	4	4	no	—
85	C₂×C₄×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	—	4	4	no	—
87	C₂×C₄×C₆₇	—	4	4	no	—
88	C₂×C₄×C₆₇	—	4	4	no	—
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	—	4	4	no	—
91	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
92	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
93	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
94	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
95	C₈×C₆₇	—	4	4	no	—
96	C₂²×C₆₇	—	2	2	no	—
97	C₂²×C₆₇	—	2	4	no	—
98	C₂²×C₆₇	—	4	1	no	—
99	C₂²×C₆₇	—	4	1	no	—
100	C₂²×C₆₇	—	4	2	no	—
101	C₂²×C₆₇	#534	1	1	yes	characteristic
102	C₂²×C₆₇	#536	1	2	yes	—
103	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
104	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
105	C₂²×C₆₇	#575	1	1	yes	characteristic
106	C₄×C₆₇	—	2	4	no	—
107	C₄×C₆₇	—	2	4	no	—
108	C₄×C₆₇	—	4	1	no	—
109	C₄×C₆₇	—	8	4	no	—
110	C₄×C₆₇	—	8	4	no	—
111	C₄×C₆₇	#563	1	4	yes	—
112	#32790	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₆₇	—	4	2	no	—
114	C₂×C₆₇	#2817	1	1	yes	characteristic
115	C₂×C₆₇	#3189	1	1	yes	characteristic
116	C₂×C₆₇	#3192	1	1	yes	characteristic
117	C₂×C₆₇	#3230	1	2	yes	—
118	C₂×C₆₇	#3235	1	2	yes	—
119	#2857	C₂×C₆₇	1	2	yes	—
120	#2100	C₂×C₆₇	1	4	yes	complemented
121	#2076	C₂×C₆₇	1	4	yes	—
122	#2057	C₂×C₆₇	1	4	yes	complemented
123	#1817	C₂×C₆₇	1	1	yes	characteristic
124	C₆₇	#32790	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
126	C₂×C₂²⋊Q₈	—	2	2	no	—
127	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
128	C₂×#107	C₂²×C₆₇	1	4	yes	—
129	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
130	#391	—	2	2	no	—
131	#389	—	2	2	no	—
132	#389	C₂²×C₆₇	1	4	yes	—
133	#384	C₂²×C₆₇	1	2	yes	—
134	#380	—	2	2	no	—
135	#380	C₂²×C₆₇	1	4	yes	—
136	#340	C₂²×C₆₇	1	8	yes	—
137	#340	C₂²×C₆₇	1	4	yes	—
138	#336	C₂²×C₆₇	1	4	yes	—
139	C₂²×Q₈	—	2	2	no	—
140	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
141	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
142	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
144	C₄⋊Q₈	—	4	4	no	—
145	C₂²⋊Q₈	—	4	8	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	4	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
151	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
152	C₂×C₂²⋊C₄	—	2	4	no	—
153	C₂×C₄²	—	4	2	no	—
154	C₂×C₄²	D₄×C₆₇	1	2	yes	—
155	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
156	#107	—	4	4	no	—
157	C₂²⋊C₈	—	4	2	no	—
158	#95	—	2	4	no	—
159	#95	—	2	4	no	—
160	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
161	C₂×Q₈	—	4	4	no	—
162	C₂×Q₈	—	4	4	no	—
163	C₂×Q₈	—	4	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	1	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
173	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
175	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
176	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	8	no	—
179	C₄⋊C₄	—	2	8	no	—
180	C₄⋊C₄	—	2	8	no	—
181	C₄⋊C₄	—	4	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	8	no	—
185	C₂²⋊C₄	—	4	8	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	2	no	—
188	C₂³	—	2	1	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	1	no	—
192	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
193	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
194	Q₈	—	8	8	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
211	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
212	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
213	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
214	C₈	—	4	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	4	1	no	—
218	C₂²	—	4	1	no	—
219	C₂²	—	4	2	no	—
220	C₂²	#534×C₆₇	1	1	yes	characteristic
221	C₂²	#536×C₆₇	1	2	yes	—
222	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
223	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
224	C₂²	#575×C₆₇	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	2	4	no	—
227	C₄	—	4	1	no	—
228	C₄	—	8	4	no	—
229	C₄	—	8	4	no	—
230	C₄	#563×C₆₇	1	4	yes	—
231	C₂	—	4	2	no	—
232	C₂	C₆₇×#2817	1	1	yes	characteristic
233	C₂	C₆₇×#3189	1	1	yes	characteristic
234	C₂	C₆₇×#3192	1	1	yes	characteristic
235	C₂	C₆₇×#3230	1	2	yes	—
236	C₂	C₆₇×#3235	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#2857

C₂

yes

maximal

C₆₇×#2100

C₂

yes

maximal, complemented

C₆₇×#2076

C₂

yes

maximal

C₆₇×#2057

C₂

yes

maximal, complemented

C₆₇×#1817

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

C₂×C₂²⋊Q₈×C₆₇

—

C₂²×C₄²×C₆₇

C₂²

yes

characteristic

C₂×#107×C₆₇

C₂²

yes

—

C₂×C₂²⋊C₈×C₆₇

C₂²

yes

—

#391×C₆₇

—

#389×C₆₇

—

#389×C₆₇

C₂²

yes

—

#384×C₆₇

C₂²

yes

—

#380×C₆₇

—

#380×C₆₇

C₂²

yes

—

#340×C₆₇

C₂²

yes

—

#340×C₆₇

C₂²

yes

—

#336×C₆₇

C₂²

yes

—

C₂²×Q₈×C₆₇

—

C₂³×C₄×C₆₇

D₄

yes

characteristic

C₂³×C₄×C₆₇

D₄

yes

characteristic

C₂³×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

—

C₄⋊Q₈×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂³

yes

—

C₂×C₄⋊C₄×C₆₇

C₂³

yes

—

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

—

#107×C₆₇

—

C₂²⋊C₈×C₆₇

—

#95×C₆₇

—

#95×C₆₇

—

C₂⁴×C₆₇

C₂×D₄

yes

characteristic

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

Q₁₆

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×Q₁₆

yes

—

C₂³×C₆₇

C₂²×D₄

yes

characteristic

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

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

H(5,𝔽₂)

yes

—

C₂×C₄×C₆₇

2₋¹⁺⁴

yes

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

100

C₂²×C₆₇

—

101

C₂²×C₆₇

#534

yes

characteristic

102

C₂²×C₆₇

#536

yes

—

103

C₂²×C₆₇

C₂²×Q₁₆

yes

characteristic

104

C₂²×C₆₇

C₂×AGL(1,ℤ/8)

yes

—

105

C₂²×C₆₇

#575

yes

characteristic

106

C₄×C₆₇

—

107

C₄×C₆₇

—

108

C₄×C₆₇

—

109

C₄×C₆₇

—

110

C₄×C₆₇

—

111

C₄×C₆₇

#563

yes

—

112

#32790

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

113

C₂×C₆₇

—

114

C₂×C₆₇

#2817

yes

characteristic

115

C₂×C₆₇

#3189

yes

characteristic

116

C₂×C₆₇

#3192

yes

characteristic

117

C₂×C₆₇

#3230

yes

—

118

C₂×C₆₇

#3235

yes

—

119

#2857

C₂×C₆₇

yes

—

120

#2100

C₂×C₆₇

yes

complemented

121

#2076

C₂×C₆₇

yes

—

122

#2057

C₂×C₆₇

yes

complemented

123

#1817

C₂×C₆₇

yes

characteristic

124

C₆₇

#32790

yes

characteristic, direct product factor, Sylow

125

C₂×C₄⋊Q₈

C₂²×C₆₇

yes

—

126

C₂×C₂²⋊Q₈

—

127

C₂²×C₄²

C₂²×C₆₇

yes

characteristic

128

C₂×#107

C₂²×C₆₇

yes

—

129

C₂×C₂²⋊C₈

C₂²×C₆₇

yes

—

130

#391

—

131

#389

—

132

#389

C₂²×C₆₇

yes

—

133

#384

C₂²×C₆₇

yes

—

134

#380

—

135

#380

C₂²×C₆₇

yes

—

136

#340

C₂²×C₆₇

yes

—

137

#340

C₂²×C₆₇

yes

—

138

#336

C₂²×C₆₇

yes

—

139

C₂²×Q₈

—

140

C₂³×C₄

D₄×C₆₇

yes

characteristic

141

C₂³×C₄

D₄×C₆₇

yes

characteristic

142

C₂³×C₄

C₂³×C₆₇

yes

characteristic

143

C₂²×C₈

C₂³×C₆₇

yes

—

144

C₄⋊Q₈

—

145

C₂²⋊Q₈

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

C₂³×C₆₇

yes

—

151

C₂×C₄⋊C₄

C₂³×C₆₇

yes

—

152

C₂×C₂²⋊C₄

—

153

C₂×C₄²

—

154

C₂×C₄²

D₄×C₆₇

yes

—

155

C₂×C₄²

C₂³×C₆₇

yes

—

156

#107

—

157

C₂²⋊C₈

—

158

#95

—

159

#95

—

160

C₂⁴

C₂×D₄×C₆₇

yes

characteristic

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂²×C₄

—

165

C₂²×C₄

—

166

C₂²×C₄

—

167

C₂²×C₄

—

168

C₂²×C₄

—

169

C₂²×C₄

—

170

C₂²×C₄

—

171

C₂²×C₄

—

172

C₂²×C₄

Q₁₆×C₆₇

yes

—

173

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

174

C₂²×C₄

C₂×D₄×C₆₇

yes

—

175

C₂²×C₄

C₂×D₄×C₆₇

yes

—

176

C₂²×C₄

C₂⁴×C₆₇

yes

characteristic

177

C₂×C₈

—

178

C₂×C₈

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₂²⋊C₄

—

186

C₄²

—

187

C₄²

—

188

C₂³

—

189

C₂³

—

190

C₂³

—

191

C₂³

—

192

C₂³

C₂×Q₁₆×C₆₇

yes

—

193

C₂³

C₂²×D₄×C₆₇

yes

characteristic

194

Q₈

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

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

C₂×Q₁₆×C₆₇

yes

—

211

C₂×C₄

AGL(1,ℤ/8)×C₆₇

yes

—

212

C₂×C₄

H(5,𝔽₂)×C₆₇

yes

—

213

C₂×C₄

2₋¹⁺⁴×C₆₇

yes

—

214

C₈

—

215

C₂²

—

216

C₂²

—

217

C₂²

—

218

C₂²

—

219

C₂²

—

220

C₂²

#534×C₆₇

yes

characteristic

221

C₂²

#536×C₆₇

yes

—

222

C₂²

C₂²×Q₁₆×C₆₇

yes

characteristic

223

C₂²

C₂×AGL(1,ℤ/8)×C₆₇

yes

—

224

C₂²

#575×C₆₇

yes

characteristic

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

#563×C₆₇

yes

—

231

C₂

—

232

C₂

C₆₇×#2817

yes

characteristic

233

C₂

C₆₇×#3189

yes

characteristic

234

C₂

C₆₇×#3192

yes

characteristic

235

C₂

C₆₇×#3230

yes

—

236

C₂

C₆₇×#3235

yes

—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₆₇

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₆₇

prime radical

C₂⁴×C₆₇

C₂×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
2. C₆₇×#2100	C₂	semidirect
4. C₆₇×#2057	C₂	semidirect
112. #32790	C₆₇	direct (N × H)
120. #2100	C₂×C₆₇	semidirect
122. #2057	C₂×C₆₇	semidirect
124. C₆₇	#32790	direct (N × H)

H ≅ G/N

type

2. C₆₇×#2100

C₂

semidirect

4. C₆₇×#2057

C₂

semidirect

112. #32790

C₆₇

direct (N × H)

120. #2100

C₂×C₆₇

semidirect

122. #2057

C₂×C₆₇

semidirect

124. C₆₇

#32790

direct (N × H)

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

subgroup	index
C₆₇×#2857	2
C₆₇×#2100	2
C₆₇×#2076	2
C₆₇×#2057	2
C₆₇×#1817	2
#32790	67

subgroup

index

C₆₇×#2857

C₆₇×#2100

C₆₇×#2076

C₆₇×#2057

C₆₇×#1817

#32790

Subgroups — C₆₇×#32790 #30712041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#32790 #30712041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32790 #30712041