Subgroups — C₂×C₆₇×#2931 #30739386

Order 17152.

2894 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2931	C₂	1	16	yes	maximal, direct product factor
4	C₂×#436×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×C₈×D₄×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#408×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	C₂²	1	2	yes	—
10	C₂×#131×C₆₇	—	2	16	no	—
11	C₂×#131×C₆₇	C₂²	1	16	yes	complemented
12	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#532×C₆₇	C₄	1	8	yes	—
18	#532×C₆₇	C₂²	1	8	yes	—
19	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
20	C₂×#127×C₆₇	C₄	1	1	yes	characteristic
21	C₂×#124×C₆₇	C₄	1	2	yes	—
22	C₂×#121×C₆₇	C₄	1	4	yes	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	#436×C₆₇	C₂²	1	32	yes	complemented
26	C₈×D₄×C₆₇	C₂²	1	32	yes	complemented
27	#423×C₆₇	C₂²	1	8	yes	complemented
28	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
29	#408×C₆₇	C₂²	1	16	yes	complemented
30	C₂×C₂²⋊C₈×C₆₇	C₂²	1	8	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	8	yes	complemented
34	C₂×#41×C₆₇	—	2	8	no	—
35	C₂×#41×C₆₇	—	2	8	no	—
36	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
37	C₂×#41×C₆₇	C₂³	1	8	yes	—
38	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
40	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
41	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	#131×C₆₇	—	2	64	no	—
46	C₂×M₁₆×C₆₇	—	2	8	no	—
47	C₂×M₁₆×C₆₇	—	2	16	no	—
48	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
49	C₂×M₁₆×C₆₇	C₂³	1	16	yes	—
50	C₂²×C₈×C₆₇	—	2	4	no	—
51	C₂²×C₈×C₆₇	—	2	16	no	—
52	C₂²×C₈×C₆₇	D₄	1	8	yes	complemented
53	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
54	C₂²×C₈×C₆₇	C₂³	1	16	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
57	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
61	#127×C₆₇	C₂×C₄	1	8	yes	—
62	#124×C₆₇	C₂×C₄	1	16	yes	—
63	#121×C₆₇	C₂×C₄	1	32	yes	—
64	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	—
65	C₄×D₄×C₆₇	C₂³	1	16	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
67	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
68	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
71	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
73	C₄⋊C₈×C₆₇	C₂³	1	16	yes	—
74	C₂²⋊C₈×C₆₇	C₂³	1	32	yes	—
75	C₄×C₈×C₆₇	C₂³	1	8	yes	—
76	C₄×C₈×C₆₇	C₂³	1	8	yes	—
77	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
78	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	#41×C₆₇	—	2	32	no	—
80	#41×C₆₇	—	2	16	no	—
81	#41×C₆₇	—	2	16	no	—
82	C₂×Q₈×C₆₇	—	2	8	no	—
83	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
84	C₂×D₄×C₆₇	—	2	16	no	—
85	C₂×D₄×C₆₇	—	2	8	no	—
86	C₂×D₄×C₆₇	—	2	16	no	—
87	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
88	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
89	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
90	C₂²×C₄×C₆₇	—	2	8	no	—
91	C₂²×C₄×C₆₇	—	2	8	no	—
92	C₂²×C₄×C₆₇	—	2	8	no	—
93	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
97	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
101	C₂²×C₄×C₆₇	#41	1	4	yes	—
102	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
103	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
105	M₁₆×C₆₇	—	2	16	no	—
106	M₁₆×C₆₇	—	2	32	no	—
107	C₂×C₈×C₆₇	—	2	8	no	—
108	C₂×C₈×C₆₇	—	2	4	no	—
109	C₂×C₈×C₆₇	—	2	8	no	—
110	C₂×C₈×C₆₇	—	2	32	no	—
111	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
112	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
113	C₂×C₈×C₆₇	C₂×D₄	1	16	yes	complemented
114	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
115	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
116	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
117	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
118	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
119	C₂²⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
120	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
121	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
122	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
123	C₄²×C₆₇	C₂⁴	1	4	yes	—
124	C₂³×C₆₇	—	2	4	no	—
125	C₂³×C₆₇	—	2	8	no	—
126	C₂³×C₆₇	—	2	16	no	—
127	C₂³×C₆₇	C₄×D₄	1	4	yes	—
128	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
129	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
130	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
131	Q₈×C₆₇	—	2	16	no	—
132	D₄×C₆₇	—	2	32	no	—
133	D₄×C₆₇	—	2	16	no	—
134	D₄×C₆₇	—	2	32	no	—
135	C₂×C₄×C₆₇	—	2	8	no	—
136	C₂×C₄×C₆₇	—	2	4	no	—
137	C₂×C₄×C₆₇	—	2	16	no	—
138	C₂×C₄×C₆₇	—	2	16	no	—
139	C₂×C₄×C₆₇	—	2	8	no	—
140	C₂×C₄×C₆₇	—	2	8	no	—
141	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
142	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
143	C₂×C₄×C₆₇	#131	1	8	yes	complemented
144	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
145	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
146	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
148	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
149	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
150	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
151	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
152	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
153	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
154	C₈×C₆₇	—	2	8	no	—
155	C₈×C₆₇	—	2	4	no	—
156	C₈×C₆₇	C₂²×D₄	1	4	yes	—
157	C₈×C₆₇	C₂²×D₄	1	4	yes	—
158	C₂²×C₆₇	—	2	8	no	—
159	C₂²×C₆₇	—	2	4	no	—
160	C₂²×C₆₇	—	2	8	no	—
161	C₂²×C₆₇	—	2	16	no	—
162	C₂²×C₆₇	—	4	16	no	—
163	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
164	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
165	C₂²×C₆₇	C₂×#131	1	4	yes	—
166	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
167	C₄×C₆₇	—	2	8	no	—
168	C₄×C₆₇	—	2	4	no	—
169	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
170	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
171	C₄×C₆₇	C₂×#131	1	8	yes	complemented
172	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
173	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
174	C₂×#2931	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂×C₆₇	—	2	4	no	—
176	C₂×C₆₇	—	4	8	no	—
177	C₂×C₆₇	#2931	1	4	yes	direct product factor
178	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
179	C₂×C₆₇	C₂²×#131	1	2	yes	—
180	C₂²×#131	C₂×C₆₇	1	2	yes	complemented
181	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
182	#2931	C₂×C₆₇	1	16	yes	direct product factor
183	C₂×#436	C₂×C₆₇	1	4	yes	complemented
184	C₂×C₈×D₄	C₂×C₆₇	1	4	yes	complemented
185	C₂×#423	C₂×C₆₇	1	1	yes	characteristic, complemented
186	C₂×#408	C₂×C₆₇	1	2	yes	complemented
187	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
188	C₆₇	C₂×#2931	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²×#41	C₂²×C₆₇	1	2	yes	—
190	C₂×#131	—	2	16	no	—
191	C₂×#131	C₂²×C₆₇	1	16	yes	complemented
192	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
193	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
194	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
195	C₂³×C₈	C₂²×C₆₇	1	4	yes	—
196	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
197	#532	C₄×C₆₇	1	8	yes	—
198	#532	C₂²×C₆₇	1	8	yes	—
199	C₂×C₄⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
200	C₂×#127	C₄×C₆₇	1	1	yes	characteristic
201	C₂×#124	C₄×C₆₇	1	2	yes	—
202	C₂×#121	C₄×C₆₇	1	4	yes	—
203	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
204	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
205	#436	C₂²×C₆₇	1	32	yes	complemented
206	C₈×D₄	C₂²×C₆₇	1	32	yes	complemented
207	#423	C₂²×C₆₇	1	8	yes	complemented
208	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
209	#408	C₂²×C₆₇	1	16	yes	complemented
210	C₂×C₂²⋊C₈	C₂²×C₆₇	1	8	yes	—
211	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
212	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
213	C₂×C₄×C₈	C₂²×C₆₇	1	8	yes	complemented
214	C₂×#41	—	2	8	no	—
215	C₂×#41	—	2	8	no	—
216	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
217	C₂×#41	C₂³×C₆₇	1	8	yes	—
218	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
219	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
220	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
221	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
222	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
223	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
224	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
225	#131	—	2	64	no	—
226	C₂×M₁₆	—	2	8	no	—
227	C₂×M₁₆	—	2	16	no	—
228	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
229	C₂×M₁₆	C₂³×C₆₇	1	16	yes	—
230	C₂²×C₈	—	2	4	no	—
231	C₂²×C₈	—	2	16	no	—
232	C₂²×C₈	D₄×C₆₇	1	8	yes	complemented
233	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
234	C₂²×C₈	C₂³×C₆₇	1	16	yes	—
235	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
236	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
237	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
238	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
239	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
240	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
241	#127	C₂×C₄×C₆₇	1	8	yes	—
242	#124	C₂×C₄×C₆₇	1	16	yes	—
243	#121	C₂×C₄×C₆₇	1	32	yes	—
244	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	—
245	C₄×D₄	C₂³×C₆₇	1	16	yes	—
246	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
247	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
248	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
249	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
250	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
251	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
252	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
253	C₄⋊C₈	C₂³×C₆₇	1	16	yes	—
254	C₂²⋊C₈	C₂³×C₆₇	1	32	yes	—
255	C₄×C₈	C₂³×C₆₇	1	8	yes	—
256	C₄×C₈	C₂³×C₆₇	1	8	yes	—
257	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
258	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
259	#41	—	2	32	no	—
260	#41	—	2	16	no	—
261	#41	—	2	16	no	—
262	C₂×Q₈	—	2	8	no	—
263	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
264	C₂×D₄	—	2	16	no	—
265	C₂×D₄	—	2	8	no	—
266	C₂×D₄	—	2	16	no	—
267	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
268	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
269	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
270	C₂²×C₄	—	2	8	no	—
271	C₂²×C₄	—	2	8	no	—
272	C₂²×C₄	—	2	8	no	—
273	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
274	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
275	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
276	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
277	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
278	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
279	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
280	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
281	C₂²×C₄	#41×C₆₇	1	4	yes	—
282	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
283	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
284	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
285	M₁₆	—	2	16	no	—
286	M₁₆	—	2	32	no	—
287	C₂×C₈	—	2	8	no	—
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	2	8	no	—
290	C₂×C₈	—	2	32	no	—
291	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
292	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
293	C₂×C₈	C₂×D₄×C₆₇	1	16	yes	complemented
294	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
295	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
296	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
297	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
298	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
299	C₂²⋊C₄	C₂²×C₄×C₆₇	1	32	yes	—
300	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
301	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
302	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
303	C₄²	C₂⁴×C₆₇	1	4	yes	—
304	C₂³	—	2	4	no	—
305	C₂³	—	2	8	no	—
306	C₂³	—	2	16	no	—
307	C₂³	C₄×D₄×C₆₇	1	4	yes	—
308	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
309	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
310	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
311	Q₈	—	2	16	no	—
312	D₄	—	2	32	no	—
313	D₄	—	2	16	no	—
314	D₄	—	2	32	no	—
315	C₂×C₄	—	2	8	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	16	no	—
318	C₂×C₄	—	2	16	no	—
319	C₂×C₄	—	2	8	no	—
320	C₂×C₄	—	2	8	no	—
321	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
322	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
323	C₂×C₄	#131×C₆₇	1	8	yes	complemented
324	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
325	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
326	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
327	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
328	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
329	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
330	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
331	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
332	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
333	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
334	C₈	—	2	8	no	—
335	C₈	—	2	4	no	—
336	C₈	C₂²×D₄×C₆₇	1	4	yes	—
337	C₈	C₂²×D₄×C₆₇	1	4	yes	—
338	C₂²	—	2	4	no	—
339	C₂²	—	2	8	no	—
340	C₂²	—	2	16	no	—
341	C₂²	—	2	8	no	—
342	C₂²	—	4	16	no	—
343	C₂²	C₂×C₄×D₄×C₆₇	1	4	yes	—
344	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
345	C₂²	C₂×#131×C₆₇	1	4	yes	—
346	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
347	C₄	—	2	4	no	—
348	C₄	—	2	8	no	—
349	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
350	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
351	C₄	C₂×#131×C₆₇	1	8	yes	complemented
352	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
353	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
354	C₂	—	2	4	no	—
355	C₂	—	4	8	no	—
356	C₂	C₆₇×#2931	1	4	yes	direct product factor
357	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
358	C₂	C₂²×#131×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₆₇	C₂³
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂×#532×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×#131×C₆₇	C₂	semidirect
3. C₆₇×#2931	C₂	direct (N × H)
4. C₂×#436×C₆₇	C₂	semidirect
5. C₂×C₈×D₄×C₆₇	C₂	semidirect
6. C₂×#423×C₆₇	C₂	semidirect
7. C₂×#408×C₆₇	C₂	semidirect
8. C₂²×C₄×C₈×C₆₇	C₂	semidirect
11. C₂×#131×C₆₇	C₂²	semidirect
25. #436×C₆₇	C₂²	semidirect
26. C₈×D₄×C₆₇	C₂²	semidirect
27. #423×C₆₇	C₂²	semidirect
29. #408×C₆₇	C₂²	semidirect
33. C₂×C₄×C₈×C₆₇	C₂²	semidirect
52. C₂²×C₈×C₆₇	D₄	semidirect
113. C₂×C₈×C₆₇	C₂×D₄	semidirect
143. C₂×C₄×C₆₇	#131	semidirect
171. C₄×C₆₇	C₂×#131	semidirect
174. C₂×#2931	C₆₇	direct (N × H)
177. C₂×C₆₇	#2931	direct (N × H)
180. C₂²×#131	C₂×C₆₇	semidirect
182. #2931	C₂×C₆₇	direct (N × H)
183. C₂×#436	C₂×C₆₇	semidirect
184. C₂×C₈×D₄	C₂×C₆₇	semidirect
185. C₂×#423	C₂×C₆₇	semidirect
186. C₂×#408	C₂×C₆₇	semidirect
187. C₂²×C₄×C₈	C₂×C₆₇	semidirect
188. C₆₇	C₂×#2931	direct (N × H)
191. C₂×#131	C₂²×C₆₇	semidirect
205. #436	C₂²×C₆₇	semidirect
206. C₈×D₄	C₂²×C₆₇	semidirect
207. #423	C₂²×C₆₇	semidirect
209. #408	C₂²×C₆₇	semidirect
213. C₂×C₄×C₈	C₂²×C₆₇	semidirect
232. C₂²×C₈	D₄×C₆₇	semidirect
293. C₂×C₈	C₂×D₄×C₆₇	semidirect
323. C₂×C₄	#131×C₆₇	semidirect
351. C₄	C₂×#131×C₆₇	semidirect
356. C₂	C₆₇×#2931	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#131×C₆₇	2
C₂×#532×C₆₇	2
C₆₇×#2931	2
C₂×#436×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#423×C₆₇	2
C₂×#408×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂×#2931	67

Subgroups — C2×C67×#2931 #30739386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2931 #30739386