Group #30692557 — subgroups

Subgroups — C₆₇×#13306 #30692557

Order 17152.

566 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 138 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	C₂²×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#467×C₆₇	C₂	1	2	yes	maximal
3	C₂×#358×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1380	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
7	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
9	C₂×Q₃₂×C₆₇	—	2	4	no	—
10	C₂×Q₃₂×C₆₇	—	2	2	no	—
11	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	#467×C₆₇	—	2	4	no	—
14	#467×C₆₇	—	2	4	no	—
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#103×C₆₇	—	2	2	no	—
17	#358×C₆₇	—	2	4	no	—
18	C₂²⋊C₁₆×C₆₇	—	2	2	no	—
19	C₂²×Q₈×C₆₇	—	2	2	no	—
20	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
21	C₂×Q₁₆×C₆₇	—	2	2	no	—
22	C₂×Q₁₆×C₆₇	—	2	2	no	—
23	C₂×Q₁₆×C₆₇	—	2	2	no	—
24	C₂×Q₁₆×C₆₇	—	2	2	no	—
25	C₂×Q₁₆×C₆₇	—	2	2	no	—
26	C₂×Q₁₆×C₆₇	D₄	1	4	yes	—
27	C₂²×C₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	—	2	1	no	—
30	C₂²×C₈×C₆₇	—	2	1	no	—
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂²⋊Q₈×C₆₇	—	2	8	no	—
34	C₂²⋊Q₈×C₆₇	—	4	4	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
37	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
38	Q₃₂×C₆₇	—	2	8	no	—
39	C₂×C₁₆×C₆₇	—	2	2	no	—
40	C₂×C₁₆×C₆₇	—	2	1	no	—
41	C₂×C₁₆×C₆₇	—	2	2	no	—
42	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
43	#107×C₆₇	—	2	2	no	—
44	#107×C₆₇	D₄	1	4	yes	—
45	#103×C₆₇	—	2	8	no	—
46	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
47	C₂×Q₈×C₆₇	—	2	4	no	—
48	C₂×Q₈×C₆₇	—	4	2	no	—
49	C₂×Q₈×C₆₇	—	4	2	no	—
50	C₂×Q₈×C₆₇	—	4	2	no	—
51	C₂×Q₈×C₆₇	—	4	2	no	—
52	C₂×Q₈×C₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	—	2	1	no	—
55	C₂²×C₄×C₆₇	—	2	1	no	—
56	C₂²×C₄×C₆₇	—	2	2	no	—
57	C₂²×C₄×C₆₇	—	4	2	no	—
58	C₂²×C₄×C₆₇	—	4	2	no	—
59	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₆₇	—	2	4	no	—
62	Q₁₆×C₆₇	—	2	4	no	—
63	Q₁₆×C₆₇	—	4	2	no	—
64	Q₁₆×C₆₇	—	4	2	no	—
65	C₂×C₈×C₆₇	—	2	2	no	—
66	C₂×C₈×C₆₇	—	2	1	no	—
67	C₂×C₈×C₆₇	—	4	1	no	—
68	C₂×C₈×C₆₇	—	4	2	no	—
69	C₂×C₈×C₆₇	—	4	1	no	—
70	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
74	C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₄⋊C₄×C₆₇	—	4	8	no	—
76	C₄⋊C₄×C₆₇	—	4	2	no	—
77	C₂²⋊C₄×C₆₇	—	4	8	no	—
78	C₁₆×C₆₇	—	2	2	no	—
79	C₁₆×C₆₇	—	4	1	no	—
80	C₂³×C₆₇	—	2	2	no	—
81	C₂³×C₆₇	—	2	1	no	—
82	C₂³×C₆₇	—	2	1	no	—
83	C₂³×C₆₇	—	2	2	no	—
84	C₂³×C₆₇	Q₃₂	1	2	yes	complemented
85	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
86	Q₈×C₆₇	—	4	4	no	—
87	Q₈×C₆₇	—	4	4	no	—
88	Q₈×C₆₇	—	8	2	no	—
89	Q₈×C₆₇	—	8	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	4	1	no	—
93	C₂×C₄×C₆₇	—	4	2	no	—
94	C₂×C₄×C₆₇	—	4	1	no	—
95	C₂×C₄×C₆₇	—	4	4	no	—
96	C₂×C₄×C₆₇	—	4	4	no	—
97	C₂×C₄×C₆₇	—	8	2	no	—
98	C₂×C₄×C₆₇	—	8	2	no	—
99	C₂×C₄×C₆₇	—	8	2	no	—
100	C₂×C₄×C₆₇	—	8	2	no	—
101	C₂×C₄×C₆₇	#120	1	2	yes	—
102	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
104	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
105	C₈×C₆₇	—	4	1	no	—
106	C₈×C₆₇	#120	1	1	yes	characteristic
107	C₈×C₆₇	#121	1	2	yes	—
108	C₈×C₆₇	#127	1	1	yes	characteristic
109	C₂²×C₆₇	—	2	4	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	4	2	no	—
112	C₂²×C₆₇	—	4	1	no	—
113	C₂²×C₆₇	—	4	1	no	—
114	C₂²×C₆₇	#447	1	2	yes	—
115	C₂²×C₆₇	#466	1	1	yes	characteristic
116	C₂²×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
117	C₂²×C₆₇	#508	1	1	yes	characteristic
118	C₂²×C₆₇	#510	1	2	yes	—
119	C₄×C₆₇	—	4	1	no	—
120	C₄×C₆₇	—	8	4	no	—
121	C₄×C₆₇	—	8	4	no	—
122	C₄×C₆₇	#392	1	1	yes	characteristic
123	C₄×C₆₇	#449	1	2	yes	—
124	C₄×C₆₇	#469	1	1	yes	characteristic
125	#13306	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×C₆₇	—	4	2	no	—
127	C₂×C₆₇	#1999	1	1	yes	characteristic
128	C₂×C₆₇	#2188	1	2	yes	—
129	C₂×C₆₇	#2189	1	2	yes	—
130	C₂×C₆₇	#2216	1	1	yes	characteristic
131	C₂×C₆₇	#2219	1	1	yes	characteristic
132	C₂²×Q₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
133	C₂×#467	C₂×C₆₇	1	2	yes	—
134	C₂×#358	C₂×C₆₇	1	2	yes	complemented
135	C₂×C₂²⋊C₁₆	C₂×C₆₇	1	1	yes	characteristic
136	#1380	C₂×C₆₇	1	1	yes	characteristic, complemented
137	C₆₇	#13306	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
139	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
140	C₂×C₂²⋊Q₈	—	2	2	no	—
141	C₂×Q₃₂	—	2	4	no	—
142	C₂×Q₃₂	—	2	2	no	—
143	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
144	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
145	#467	—	2	4	no	—
146	#467	—	2	4	no	—
147	C₂×#107	C₂²×C₆₇	1	2	yes	—
148	C₂×#103	—	2	2	no	—
149	#358	—	2	4	no	—
150	C₂²⋊C₁₆	—	2	2	no	—
151	C₂²×Q₈	—	2	2	no	—
152	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	2	2	no	—
155	C₂×Q₁₆	—	2	2	no	—
156	C₂×Q₁₆	—	2	2	no	—
157	C₂×Q₁₆	—	2	2	no	—
158	C₂×Q₁₆	D₄×C₆₇	1	4	yes	—
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
165	C₂²⋊Q₈	—	2	8	no	—
166	C₂²⋊Q₈	—	4	4	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	4	2	no	—
169	C₂×C₂²⋊C₄	—	4	2	no	—
170	Q₃₂	—	2	8	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	1	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	D₄×C₆₇	1	2	yes	—
175	#107	—	2	2	no	—
176	#107	D₄×C₆₇	1	4	yes	—
177	#103	—	2	8	no	—
178	C₂⁴	D₈×C₆₇	1	1	yes	characteristic
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
193	Q₁₆	—	2	4	no	—
194	Q₁₆	—	2	4	no	—
195	Q₁₆	—	4	2	no	—
196	Q₁₆	—	4	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
205	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	8	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₂²⋊C₄	—	4	8	no	—
210	C₁₆	—	2	2	no	—
211	C₁₆	—	4	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	Q₃₂×C₆₇	1	2	yes	complemented
217	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
218	Q₈	—	4	4	no	—
219	Q₈	—	4	4	no	—
220	Q₈	—	8	2	no	—
221	Q₈	—	8	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	#120×C₆₇	1	2	yes	—
234	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
235	C₂×C₄	#135×C₆₇	1	1	yes	characteristic
236	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
237	C₈	—	4	1	no	—
238	C₈	#120×C₆₇	1	1	yes	characteristic
239	C₈	#121×C₆₇	1	2	yes	—
240	C₈	#127×C₆₇	1	1	yes	characteristic
241	C₂²	—	2	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	1	no	—
246	C₂²	#447×C₆₇	1	2	yes	—
247	C₂²	#466×C₆₇	1	1	yes	characteristic
248	C₂²	C₂×Q₃₂×C₆₇	1	1	yes	characteristic
249	C₂²	#508×C₆₇	1	1	yes	characteristic
250	C₂²	#510×C₆₇	1	2	yes	—
251	C₄	—	4	1	no	—
252	C₄	—	8	4	no	—
253	C₄	—	8	4	no	—
254	C₄	#392×C₆₇	1	1	yes	characteristic
255	C₄	#449×C₆₇	1	2	yes	—
256	C₄	#469×C₆₇	1	1	yes	characteristic
257	C₂	—	4	2	no	—
258	C₂	C₆₇×#1999	1	1	yes	characteristic
259	C₂	C₆₇×#2188	1	2	yes	—
260	C₂	C₆₇×#2189	1	2	yes	—
261	C₂	C₆₇×#2216	1	1	yes	characteristic
262	C₂	C₆₇×#2219	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#467×C₆₇

C₂

yes

maximal

C₂×#358×C₆₇

C₂

yes

maximal, complemented

C₂×C₂²⋊C₁₆×C₆₇

C₂

yes

maximal, characteristic

C₆₇×#1380

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₆₇

C₂²

yes

—

C₂³×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₂²⋊Q₈×C₆₇

—

C₂×Q₃₂×C₆₇

—

C₂×Q₃₂×C₆₇

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

#467×C₆₇

—

#467×C₆₇

—

C₂×#107×C₆₇

C₂²

yes

—

C₂×#103×C₆₇

—

#358×C₆₇

—

C₂²⋊C₁₆×C₆₇

—

C₂²×Q₈×C₆₇

—

C₂³×C₄×C₆₇

D₄

yes

characteristic

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂²⋊Q₈×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₂²⋊C₄×C₆₇

—

Q₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#107×C₆₇

—

#107×C₆₇

D₄

yes

—

#103×C₆₇

—

C₂⁴×C₆₇

D₈

yes

characteristic

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×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

Q₃₂

yes

complemented

C₂³×C₆₇

C₂×D₈

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

—

100

C₂×C₄×C₆₇

—

101

C₂×C₄×C₆₇

#120

yes

—

102

C₂×C₄×C₆₇

#121

yes

characteristic

103

C₂×C₄×C₆₇

#135

yes

characteristic

104

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

105

C₈×C₆₇

—

106

C₈×C₆₇

#120

yes

characteristic

107

C₈×C₆₇

#121

yes

—

108

C₈×C₆₇

#127

yes

characteristic

109

C₂²×C₆₇

—

110

C₂²×C₆₇

—

111

C₂²×C₆₇

—

112

C₂²×C₆₇

—

113

C₂²×C₆₇

—

114

C₂²×C₆₇

#447

yes

—

115

C₂²×C₆₇

#466

yes

characteristic

116

C₂²×C₆₇

C₂×Q₃₂

yes

characteristic

117

C₂²×C₆₇

#508

yes

characteristic

118

C₂²×C₆₇

#510

yes

—

119

C₄×C₆₇

—

120

C₄×C₆₇

—

121

C₄×C₆₇

—

122

C₄×C₆₇

#392

yes

characteristic

123

C₄×C₆₇

#449

yes

—

124

C₄×C₆₇

#469

yes

characteristic

125

#13306

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

126

C₂×C₆₇

—

127

C₂×C₆₇

#1999

yes

characteristic

128

C₂×C₆₇

#2188

yes

—

129

C₂×C₆₇

#2189

yes

—

130

C₂×C₆₇

#2216

yes

characteristic

131

C₂×C₆₇

#2219

yes

characteristic

132

C₂²×Q₃₂

C₂×C₆₇

yes

characteristic, complemented

133

C₂×#467

C₂×C₆₇

yes

—

134

C₂×#358

C₂×C₆₇

yes

complemented

135

C₂×C₂²⋊C₁₆

C₂×C₆₇

yes

characteristic

136

#1380

C₂×C₆₇

yes

characteristic, complemented

137

C₆₇

#13306

yes

characteristic, direct product factor, Sylow

138

C₂²×Q₁₆

C₂²×C₆₇

yes

—

139

C₂³×C₈

C₂²×C₆₇

yes

characteristic

140

C₂×C₂²⋊Q₈

—

141

C₂×Q₃₂

—

142

C₂×Q₃₂

—

143

C₂²×C₁₆

C₂²×C₆₇

yes

characteristic

144

C₂²×C₁₆

C₂²×C₆₇

yes

characteristic

145

#467

—

146

#467

—

147

C₂×#107

C₂²×C₆₇

yes

—

148

C₂×#103

—

149

#358

—

150

C₂²⋊C₁₆

—

151

C₂²×Q₈

—

152

C₂³×C₄

D₄×C₆₇

yes

characteristic

153

C₂×Q₁₆

—

154

C₂×Q₁₆

—

155

C₂×Q₁₆

—

156

C₂×Q₁₆

—

157

C₂×Q₁₆

—

158

C₂×Q₁₆

D₄×C₆₇

yes

—

159

C₂²×C₈

—

160

C₂²×C₈

—

161

C₂²×C₈

—

162

C₂²×C₈

—

163

C₂²×C₈

D₄×C₆₇

yes

characteristic

164

C₂²×C₈

C₂³×C₆₇

yes

characteristic

165

C₂²⋊Q₈

—

166

C₂²⋊Q₈

—

167

C₂×C₄⋊C₄

—

168

C₂×C₄⋊C₄

—

169

C₂×C₂²⋊C₄

—

170

Q₃₂

—

171

C₂×C₁₆

—

172

C₂×C₁₆

—

173

C₂×C₁₆

—

174

C₂×C₁₆

D₄×C₆₇

yes

—

175

#107

—

176

#107

D₄×C₆₇

yes

—

177

#103

—

178

C₂⁴

D₈×C₆₇

yes

characteristic

179

C₂×Q₈

—

180

C₂×Q₈

—

181

C₂×Q₈

—

182

C₂×Q₈

—

183

C₂×Q₈

—

184

C₂×Q₈

—

185

C₂²×C₄

—

186

C₂²×C₄

—

187

C₂²×C₄

—

188

C₂²×C₄

—

189

C₂²×C₄

—

190

C₂²×C₄

—

191

C₂²×C₄

D₈×C₆₇

yes

characteristic

192

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

193

Q₁₆

—

194

Q₁₆

—

195

Q₁₆

—

196

Q₁₆

—

197

C₂×C₈

—

198

C₂×C₈

—

199

C₂×C₈

—

200

C₂×C₈

—

201

C₂×C₈

—

202

C₂×C₈

C₂×D₄×C₆₇

yes

—

203

C₂×C₈

C₂×D₄×C₆₇

yes

characteristic

204

C₂×C₈

C₂×D₄×C₆₇

yes

—

205

C₂×C₈

#41×C₆₇

yes

characteristic

206

C₄⋊C₄

—

207

C₄⋊C₄

—

208

C₄⋊C₄

—

209

C₂²⋊C₄

—

210

C₁₆

—

211

C₁₆

—

212

C₂³

—

213

C₂³

—

214

C₂³

—

215

C₂³

—

216

C₂³

Q₃₂×C₆₇

yes

complemented

217

C₂³

C₂×D₈×C₆₇

yes

characteristic

218

Q₈

—

219

Q₈

—

220

Q₈

—

221

Q₈

—

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

—

231

C₂×C₄

—

232

C₂×C₄

—

233

C₂×C₄

#120×C₆₇

yes

—

234

C₂×C₄

#121×C₆₇

yes

characteristic

235

C₂×C₄

#135×C₆₇

yes

characteristic

236

C₂×C₄

AGL(1,ℤ/8)×C₆₇

yes

—

237

C₈

—

238

C₈

#120×C₆₇

yes

characteristic

239

C₈

#121×C₆₇

yes

—

240

C₈

#127×C₆₇

yes

characteristic

241

C₂²

—

242

C₂²

—

243

C₂²

—

244

C₂²

—

245

C₂²

—

246

C₂²

#447×C₆₇

yes

—

247

C₂²

#466×C₆₇

yes

characteristic

248

C₂²

C₂×Q₃₂×C₆₇

yes

characteristic

249

C₂²

#508×C₆₇

yes

characteristic

250

C₂²

#510×C₆₇

yes

—

251

C₄

—

252

C₄

—

253

C₄

—

254

C₄

#392×C₆₇

yes

characteristic

255

C₄

#449×C₆₇

yes

—

256

C₄

#469×C₆₇

yes

characteristic

257

C₂

—

258

C₂

C₆₇×#1999

yes

characteristic

259

C₂

C₆₇×#2188

yes

—

260

C₂

C₆₇×#2189

yes

—

261

C₂

C₆₇×#2216

yes

characteristic

262

C₂

C₆₇×#2219

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₇
prime radical	C₂⁴×C₆₇	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₆₇

C₂×D₈

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₆₇

prime radical

C₂⁴×C₆₇

D₈

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

N	H ≅ G/N	type
1. C₂²×Q₃₂×C₆₇	C₂	semidirect
3. C₂×#358×C₆₇	C₂	semidirect
5. C₆₇×#1380	C₂	semidirect
84. C₂³×C₆₇	Q₃₂	semidirect
125. #13306	C₆₇	direct (N × H)
132. C₂²×Q₃₂	C₂×C₆₇	semidirect
134. C₂×#358	C₂×C₆₇	semidirect
136. #1380	C₂×C₆₇	semidirect
137. C₆₇	#13306	direct (N × H)
216. C₂³	Q₃₂×C₆₇	semidirect

H ≅ G/N

type

1. C₂²×Q₃₂×C₆₇

C₂

semidirect

3. C₂×#358×C₆₇

C₂

semidirect

5. C₆₇×#1380

C₂

semidirect

84. C₂³×C₆₇

Q₃₂

semidirect

125. #13306

C₆₇

direct (N × H)

132. C₂²×Q₃₂

C₂×C₆₇

semidirect

134. C₂×#358

C₂×C₆₇

semidirect

136. #1380

C₂×C₆₇

semidirect

137. C₆₇

#13306

direct (N × H)

216. C₂³

Q₃₂×C₆₇

semidirect

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

subgroup	index
C₂²×Q₃₂×C₆₇	2
C₂×#467×C₆₇	2
C₂×#358×C₆₇	2
C₂×C₂²⋊C₁₆×C₆₇	2
C₆₇×#1380	2
#13306	67

subgroup

index

C₂²×Q₃₂×C₆₇

C₂×#467×C₆₇

C₂×#358×C₆₇

C₂×C₂²⋊C₁₆×C₆₇

C₆₇×#1380

#13306

Subgroups — C₆₇×#13306 #30692557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#13306 #30692557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#13306 #30692557