Subgroups — C₂×C₆₇×#3498 #30742046

Order 17152.

12982 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3498	C₂	1	32	yes	maximal, direct product factor
3	C₂×#550×C₆₇	C₂	1	4	yes	maximal, complemented
4	C₂×D₄²×C₆₇	C₂	1	12	yes	maximal, complemented
5	C₂×#535×C₆₇	C₂	1	6	yes	maximal, complemented
6	C₂×#532×C₆₇	C₂	1	3	yes	maximal, complemented
7	C₂²×#127×C₆₇	C₂	1	3	yes	maximal, complemented
8	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×H(5,𝔽₂)×C₆₇	—	2	32	no	—
10	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	32	yes	complemented
11	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×#41×C₆₇	C₂²	1	6	yes	complemented
13	C₂²×#41×C₆₇	C₂²	1	4	yes	complemented
14	C₂³×D₄×C₆₇	C₂²	1	6	yes	complemented
15	C₂³×D₄×C₆₇	C₂²	1	12	yes	complemented
16	C₂³×D₄×C₆₇	C₂²	1	12	yes	complemented
17	#550×C₆₇	C₂²	1	64	yes	complemented
18	D₄²×C₆₇	C₂²	1	192	yes	complemented
19	#535×C₆₇	C₂²	1	96	yes	complemented
20	#532×C₆₇	C₂²	1	48	yes	complemented
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
22	C₂×#127×C₆₇	C₂²	1	3	yes	complemented
23	C₂×#127×C₆₇	C₂²	1	6	yes	complemented
24	C₂×#127×C₆₇	C₂²	1	12	yes	complemented
25	C₂×#127×C₆₇	C₂²	1	48	yes	complemented
26	C₂×#124×C₆₇	C₂²	1	6	yes	complemented
27	C₂×#121×C₆₇	C₂²	1	12	yes	complemented
28	C₂×#121×C₆₇	C₂²	1	24	yes	complemented
29	C₂×#120×C₆₇	C₂²	1	24	yes	complemented
30	C₄×#41×C₆₇	C₂²	1	16	yes	complemented
31	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
32	C₂×C₄×D₄×C₆₇	C₂²	1	6	yes	complemented
33	C₂×C₄×D₄×C₆₇	C₂²	1	12	yes	complemented
34	C₂×#117×C₆₇	C₂²	1	3	yes	complemented
35	C₂²×C₄²×C₆₇	C₂²	1	3	yes	complemented
36	C₂⁵×C₆₇	C₂³	1	12	yes	—
37	H(5,𝔽₂)×C₆₇	—	2	256	no	—
38	C₂×#41×C₆₇	—	2	32	no	—
39	C₂×#41×C₆₇	—	2	48	no	—
40	C₂×#41×C₆₇	D₄	1	16	yes	complemented
41	C₂×#41×C₆₇	C₂³	1	8	yes	—
42	C₂×#41×C₆₇	C₂³	1	48	yes	complemented
43	C₂×#41×C₆₇	C₂³	1	32	yes	complemented
44	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
45	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₆₇	—	2	96	no	—
47	C₂²×D₄×C₆₇	—	2	48	no	—
48	C₂²×D₄×C₆₇	—	4	48	no	—
49	C₂²×D₄×C₆₇	C₂³	1	12	yes	—
50	C₂²×D₄×C₆₇	C₂³	1	12	yes	—
51	C₂²×D₄×C₆₇	C₂³	1	12	yes	—
52	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
53	C₂²×D₄×C₆₇	C₂³	1	24	yes	—
54	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
55	C₂²×D₄×C₆₇	C₂³	1	48	yes	complemented
56	C₂²×D₄×C₆₇	C₂³	1	96	yes	complemented
57	C₂²×D₄×C₆₇	C₂³	1	96	yes	complemented
58	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	12	yes	—
60	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
61	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
62	#127×C₆₇	C₂³	1	24	yes	complemented
63	#127×C₆₇	C₂³	1	48	yes	complemented
64	#127×C₆₇	C₂³	1	96	yes	complemented
65	#124×C₆₇	C₂³	1	48	yes	complemented
66	#121×C₆₇	C₂³	1	96	yes	complemented
67	#121×C₆₇	C₂³	1	192	yes	complemented
68	#120×C₆₇	C₂³	1	192	yes	complemented
69	C₄×Q₈×C₆₇	C₂³	1	16	yes	complemented
70	C₄×D₄×C₆₇	C₂³	1	48	yes	complemented
71	C₄×D₄×C₆₇	C₂³	1	96	yes	complemented
72	#117×C₆₇	C₂³	1	24	yes	complemented
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
75	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
76	C₂×C₂²⋊C₄×C₆₇	C₂³	1	24	yes	—
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
79	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
80	C₂×C₄²×C₆₇	C₂³	1	24	yes	complemented
81	C₂⁴×C₆₇	—	2	48	no	—
82	C₂⁴×C₆₇	—	4	48	no	—
83	C₂⁴×C₆₇	C₂⁴	1	12	yes	—
84	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
85	C₂⁴×C₆₇	C₂⁴	1	48	yes	—
86	#41×C₆₇	—	2	128	no	—
87	#41×C₆₇	—	2	192	no	—
88	#41×C₆₇	C₂×D₄	1	64	yes	complemented
89	C₂×Q₈×C₆₇	—	2	8	no	—
90	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	complemented
91	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
92	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
93	C₂×D₄×C₆₇	—	2	48	no	—
94	C₂×D₄×C₆₇	—	2	384	no	—
95	C₂×D₄×C₆₇	—	2	48	no	—
96	C₂×D₄×C₆₇	—	2	24	no	—
97	C₂×D₄×C₆₇	—	2	192	no	—
98	C₂×D₄×C₆₇	—	2	48	no	—
99	C₂×D₄×C₆₇	—	4	192	no	—
100	C₂×D₄×C₆₇	—	4	48	no	—
101	C₂×D₄×C₆₇	C₂×D₄	1	24	yes	complemented
102	C₂×D₄×C₆₇	C₂⁴	1	48	yes	—
103	C₂×D₄×C₆₇	C₂⁴	1	48	yes	—
104	C₂×D₄×C₆₇	C₂⁴	1	48	yes	—
105	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
106	C₂×D₄×C₆₇	C₂⁴	1	96	yes	—
107	C₂×D₄×C₆₇	C₂⁴	1	12	yes	—
108	C₂²×C₄×C₆₇	—	2	48	no	—
109	C₂²×C₄×C₆₇	—	2	24	no	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	24	yes	complemented
111	C₂²×C₄×C₆₇	C₂⁴	1	24	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	48	yes	—
113	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
115	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
118	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
119	C₄⋊C₄×C₆₇	C₂⁴	1	24	yes	—
120	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
121	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	—
122	C₂²⋊C₄×C₆₇	C₂⁴	1	96	yes	—
123	C₄²×C₆₇	C₂⁴	1	4	yes	—
124	C₄²×C₆₇	C₂⁴	1	12	yes	—
125	C₄²×C₆₇	C₂⁴	1	24	yes	—
126	C₂³×C₆₇	—	2	12	no	—
127	C₂³×C₆₇	—	2	24	no	—
128	C₂³×C₆₇	—	2	48	no	—
129	C₂³×C₆₇	—	2	96	no	—
130	C₂³×C₆₇	—	4	96	no	—
131	C₂³×C₆₇	—	8	96	no	—
132	C₂³×C₆₇	C₂²×D₄	1	12	yes	—
133	C₂³×C₆₇	C₂⁵	1	24	yes	—
134	C₂³×C₆₇	C₂⁵	1	6	yes	—
135	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
136	Q₈×C₆₇	—	2	16	no	—
137	Q₈×C₆₇	C₂²×D₄	1	16	yes	complemented
138	D₄×C₆₇	—	2	96	no	—
139	D₄×C₆₇	—	2	96	no	—
140	D₄×C₆₇	—	2	48	no	—
141	D₄×C₆₇	—	2	96	no	—
142	D₄×C₆₇	—	4	96	no	—
143	D₄×C₆₇	C₂²×D₄	1	48	yes	complemented
144	C₂×C₄×C₆₇	—	2	8	no	—
145	C₂×C₄×C₆₇	—	2	12	no	—
146	C₂×C₄×C₆₇	—	2	96	no	—
147	C₂×C₄×C₆₇	—	2	48	no	—
148	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
149	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
150	C₂×C₄×C₆₇	C₂²×D₄	1	48	yes	complemented
151	C₂×C₄×C₆₇	H(5,𝔽₂)	1	8	yes	complemented
152	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
153	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
154	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
155	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
156	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
157	C₂²×C₆₇	—	2	12	no	—
158	C₂²×C₆₇	—	2	24	no	—
159	C₂²×C₆₇	—	2	24	no	—
160	C₂²×C₆₇	—	2	48	no	—
161	C₂²×C₆₇	—	4	48	no	—
162	C₂²×C₆₇	—	8	96	no	—
163	C₂²×C₆₇	C₂³×D₄	1	12	yes	—
164	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
165	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
166	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
167	C₄×C₆₇	—	2	8	no	—
168	C₄×C₆₇	—	2	12	no	—
169	C₄×C₆₇	C₂³×D₄	1	12	yes	—
170	C₄×C₆₇	C₂³×D₄	1	4	yes	—
171	C₄×C₆₇	C₂×H(5,𝔽₂)	1	8	yes	complemented
172	C₂×#3498	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂×C₆₇	—	2	12	no	—
174	C₂×C₆₇	—	4	24	no	—
175	C₂×C₆₇	#3498	1	4	yes	direct product factor
176	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
177	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	2	yes	—
178	C₂²×H(5,𝔽₂)	C₂×C₆₇	1	2	yes	complemented
179	#3498	C₂×C₆₇	1	32	yes	direct product factor
180	C₂×#550	C₂×C₆₇	1	4	yes	complemented
181	C₂×D₄²	C₂×C₆₇	1	12	yes	complemented
182	C₂×#535	C₂×C₆₇	1	6	yes	complemented
183	C₂×#532	C₂×C₆₇	1	3	yes	complemented
184	C₂²×#127	C₂×C₆₇	1	3	yes	complemented
185	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
186	C₆₇	C₂×#3498	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×H(5,𝔽₂)	—	2	32	no	—
188	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	32	yes	complemented
189	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
190	C₂²×#41	C₂²×C₆₇	1	6	yes	complemented
191	C₂²×#41	C₂²×C₆₇	1	4	yes	complemented
192	C₂³×D₄	C₂²×C₆₇	1	6	yes	complemented
193	C₂³×D₄	C₂²×C₆₇	1	12	yes	complemented
194	C₂³×D₄	C₂²×C₆₇	1	12	yes	complemented
195	#550	C₂²×C₆₇	1	64	yes	complemented
196	D₄²	C₂²×C₆₇	1	192	yes	complemented
197	#535	C₂²×C₆₇	1	96	yes	complemented
198	#532	C₂²×C₆₇	1	48	yes	complemented
199	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
200	C₂×#127	C₂²×C₆₇	1	3	yes	complemented
201	C₂×#127	C₂²×C₆₇	1	6	yes	complemented
202	C₂×#127	C₂²×C₆₇	1	12	yes	complemented
203	C₂×#127	C₂²×C₆₇	1	48	yes	complemented
204	C₂×#124	C₂²×C₆₇	1	6	yes	complemented
205	C₂×#121	C₂²×C₆₇	1	12	yes	complemented
206	C₂×#121	C₂²×C₆₇	1	24	yes	complemented
207	C₂×#120	C₂²×C₆₇	1	24	yes	complemented
208	C₄×#41	C₂²×C₆₇	1	16	yes	complemented
209	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
210	C₂×C₄×D₄	C₂²×C₆₇	1	6	yes	complemented
211	C₂×C₄×D₄	C₂²×C₆₇	1	12	yes	complemented
212	C₂×#117	C₂²×C₆₇	1	3	yes	complemented
213	C₂²×C₄²	C₂²×C₆₇	1	3	yes	complemented
214	C₂⁵	C₂³×C₆₇	1	12	yes	—
215	H(5,𝔽₂)	—	2	256	no	—
216	C₂×#41	—	2	32	no	—
217	C₂×#41	—	2	48	no	—
218	C₂×#41	D₄×C₆₇	1	16	yes	complemented
219	C₂×#41	C₂³×C₆₇	1	8	yes	—
220	C₂×#41	C₂³×C₆₇	1	48	yes	complemented
221	C₂×#41	C₂³×C₆₇	1	32	yes	complemented
222	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
223	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
224	C₂²×D₄	—	2	96	no	—
225	C₂²×D₄	—	2	48	no	—
226	C₂²×D₄	—	4	48	no	—
227	C₂²×D₄	C₂³×C₆₇	1	12	yes	—
228	C₂²×D₄	C₂³×C₆₇	1	12	yes	—
229	C₂²×D₄	C₂³×C₆₇	1	12	yes	—
230	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
231	C₂²×D₄	C₂³×C₆₇	1	24	yes	—
232	C₂²×D₄	C₂³×C₆₇	1	3	yes	—
233	C₂²×D₄	C₂³×C₆₇	1	48	yes	complemented
234	C₂²×D₄	C₂³×C₆₇	1	96	yes	complemented
235	C₂²×D₄	C₂³×C₆₇	1	96	yes	complemented
236	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
237	C₂³×C₄	C₂³×C₆₇	1	12	yes	—
238	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
239	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
240	#127	C₂³×C₆₇	1	24	yes	complemented
241	#127	C₂³×C₆₇	1	48	yes	complemented
242	#127	C₂³×C₆₇	1	96	yes	complemented
243	#124	C₂³×C₆₇	1	48	yes	complemented
244	#121	C₂³×C₆₇	1	96	yes	complemented
245	#121	C₂³×C₆₇	1	192	yes	complemented
246	#120	C₂³×C₆₇	1	192	yes	complemented
247	C₄×Q₈	C₂³×C₆₇	1	16	yes	complemented
248	C₄×D₄	C₂³×C₆₇	1	48	yes	complemented
249	C₄×D₄	C₂³×C₆₇	1	96	yes	complemented
250	#117	C₂³×C₆₇	1	24	yes	complemented
251	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
252	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
253	C₂×C₂²⋊C₄	C₂³×C₆₇	1	6	yes	—
254	C₂×C₂²⋊C₄	C₂³×C₆₇	1	24	yes	—
255	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
256	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
257	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
258	C₂×C₄²	C₂³×C₆₇	1	24	yes	complemented
259	C₂⁴	—	2	48	no	—
260	C₂⁴	—	4	48	no	—
261	C₂⁴	C₂⁴×C₆₇	1	12	yes	—
262	C₂⁴	C₂⁴×C₆₇	1	3	yes	—
263	C₂⁴	C₂⁴×C₆₇	1	48	yes	—
264	#41	—	2	128	no	—
265	#41	—	2	192	no	—
266	#41	C₂×D₄×C₆₇	1	64	yes	complemented
267	C₂×Q₈	—	2	8	no	—
268	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	complemented
269	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
270	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
271	C₂×D₄	—	2	48	no	—
272	C₂×D₄	—	2	384	no	—
273	C₂×D₄	—	2	48	no	—
274	C₂×D₄	—	2	24	no	—
275	C₂×D₄	—	2	192	no	—
276	C₂×D₄	—	2	48	no	—
277	C₂×D₄	—	4	192	no	—
278	C₂×D₄	—	4	48	no	—
279	C₂×D₄	C₂×D₄×C₆₇	1	24	yes	complemented
280	C₂×D₄	C₂⁴×C₆₇	1	48	yes	—
281	C₂×D₄	C₂⁴×C₆₇	1	48	yes	—
282	C₂×D₄	C₂⁴×C₆₇	1	48	yes	—
283	C₂×D₄	C₂⁴×C₆₇	1	24	yes	—
284	C₂×D₄	C₂⁴×C₆₇	1	96	yes	—
285	C₂×D₄	C₂⁴×C₆₇	1	12	yes	—
286	C₂²×C₄	—	2	48	no	—
287	C₂²×C₄	—	2	24	no	—
288	C₂²×C₄	C₂×D₄×C₆₇	1	24	yes	complemented
289	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
290	C₂²×C₄	C₂⁴×C₆₇	1	24	yes	—
291	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
292	C₂²×C₄	C₂⁴×C₆₇	1	48	yes	—
293	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
295	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
296	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
297	C₄⋊C₄	C₂⁴×C₆₇	1	24	yes	—
298	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
299	C₂²⋊C₄	C₂⁴×C₆₇	1	24	yes	—
300	C₂²⋊C₄	C₂⁴×C₆₇	1	96	yes	—
301	C₄²	C₂⁴×C₆₇	1	4	yes	—
302	C₄²	C₂⁴×C₆₇	1	12	yes	—
303	C₄²	C₂⁴×C₆₇	1	24	yes	—
304	C₂³	—	2	12	no	—
305	C₂³	—	2	24	no	—
306	C₂³	—	2	48	no	—
307	C₂³	—	2	96	no	—
308	C₂³	—	4	96	no	—
309	C₂³	—	8	96	no	—
310	C₂³	C₂²×D₄×C₆₇	1	12	yes	—
311	C₂³	C₂⁵×C₆₇	1	24	yes	—
312	C₂³	C₂⁵×C₆₇	1	6	yes	—
313	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
314	Q₈	—	2	16	no	—
315	Q₈	C₂²×D₄×C₆₇	1	16	yes	complemented
316	D₄	—	2	96	no	—
317	D₄	—	2	96	no	—
318	D₄	—	2	48	no	—
319	D₄	—	2	96	no	—
320	D₄	—	4	96	no	—
321	D₄	C₂²×D₄×C₆₇	1	48	yes	complemented
322	C₂×C₄	—	2	8	no	—
323	C₂×C₄	—	2	12	no	—
324	C₂×C₄	—	2	96	no	—
325	C₂×C₄	—	2	48	no	—
326	C₂×C₄	C₂²×D₄×C₆₇	1	12	yes	—
327	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
328	C₂×C₄	C₂²×D₄×C₆₇	1	48	yes	complemented
329	C₂×C₄	H(5,𝔽₂)×C₆₇	1	8	yes	complemented
330	C₂×C₄	C₂⁵×C₆₇	1	12	yes	—
331	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
332	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
333	C₂×C₄	C₂⁵×C₆₇	1	6	yes	—
334	C₂×C₄	C₂⁵×C₆₇	1	4	yes	—
335	C₂²	—	2	12	no	—
336	C₂²	—	2	24	no	—
337	C₂²	—	2	24	no	—
338	C₂²	—	2	48	no	—
339	C₂²	—	4	48	no	—
340	C₂²	—	8	96	no	—
341	C₂²	C₂³×D₄×C₆₇	1	12	yes	—
342	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
343	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	4	yes	—
344	C₂²	C₂⁶×C₆₇	1	1	yes	characteristic
345	C₄	—	2	8	no	—
346	C₄	—	2	12	no	—
347	C₄	C₂³×D₄×C₆₇	1	12	yes	—
348	C₄	C₂³×D₄×C₆₇	1	4	yes	—
349	C₄	C₂×H(5,𝔽₂)×C₆₇	1	8	yes	complemented
350	C₂	—	2	12	no	—
351	C₂	—	4	24	no	—
352	C₂	C₆₇×#3498	1	4	yes	direct product factor
353	C₂	C₂⁴×D₄×C₆₇	1	1	yes	characteristic
354	C₂	C₂²×H(5,𝔽₂)×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×H(5,𝔽₂)×C₆₇	C₂	semidirect
2. C₆₇×#3498	C₂	direct (N × H)
3. C₂×#550×C₆₇	C₂	semidirect
4. C₂×D₄²×C₆₇	C₂	semidirect
5. C₂×#535×C₆₇	C₂	semidirect
6. C₂×#532×C₆₇	C₂	semidirect
7. C₂²×#127×C₆₇	C₂	semidirect
8. C₂×C₄×#41×C₆₇	C₂	semidirect
10. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
12. C₂²×#41×C₆₇	C₂²	semidirect
13. C₂²×#41×C₆₇	C₂²	semidirect
14. C₂³×D₄×C₆₇	C₂²	semidirect
15. C₂³×D₄×C₆₇	C₂²	semidirect
16. C₂³×D₄×C₆₇	C₂²	semidirect
17. #550×C₆₇	C₂²	semidirect
18. D₄²×C₆₇	C₂²	semidirect
19. #535×C₆₇	C₂²	semidirect
20. #532×C₆₇	C₂²	semidirect
21. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
22. C₂×#127×C₆₇	C₂²	semidirect
23. C₂×#127×C₆₇	C₂²	semidirect
24. C₂×#127×C₆₇	C₂²	semidirect
25. C₂×#127×C₆₇	C₂²	semidirect
26. C₂×#124×C₆₇	C₂²	semidirect
27. C₂×#121×C₆₇	C₂²	semidirect
28. C₂×#121×C₆₇	C₂²	semidirect
29. C₂×#120×C₆₇	C₂²	semidirect
30. C₄×#41×C₆₇	C₂²	semidirect
31. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
32. C₂×C₄×D₄×C₆₇	C₂²	semidirect
33. C₂×C₄×D₄×C₆₇	C₂²	semidirect
34. C₂×#117×C₆₇	C₂²	semidirect
35. C₂²×C₄²×C₆₇	C₂²	semidirect
40. C₂×#41×C₆₇	D₄	semidirect
42. C₂×#41×C₆₇	C₂³	semidirect
43. C₂×#41×C₆₇	C₂³	semidirect
55. C₂²×D₄×C₆₇	C₂³	semidirect
56. C₂²×D₄×C₆₇	C₂³	semidirect
57. C₂²×D₄×C₆₇	C₂³	semidirect
61. C₄⋊Q₈×C₆₇	C₂³	semidirect
62. #127×C₆₇	C₂³	semidirect
63. #127×C₆₇	C₂³	semidirect
64. #127×C₆₇	C₂³	semidirect
65. #124×C₆₇	C₂³	semidirect
66. #121×C₆₇	C₂³	semidirect
67. #121×C₆₇	C₂³	semidirect
68. #120×C₆₇	C₂³	semidirect
69. C₄×Q₈×C₆₇	C₂³	semidirect
70. C₄×D₄×C₆₇	C₂³	semidirect
71. C₄×D₄×C₆₇	C₂³	semidirect
72. #117×C₆₇	C₂³	semidirect
80. C₂×C₄²×C₆₇	C₂³	semidirect
88. #41×C₆₇	C₂×D₄	semidirect
90. C₂×Q₈×C₆₇	C₂×D₄	semidirect
101. C₂×D₄×C₆₇	C₂×D₄	semidirect
110. C₂²×C₄×C₆₇	C₂×D₄	semidirect
137. Q₈×C₆₇	C₂²×D₄	semidirect
143. D₄×C₆₇	C₂²×D₄	semidirect
150. C₂×C₄×C₆₇	C₂²×D₄	semidirect
151. C₂×C₄×C₆₇	H(5,𝔽₂)	semidirect
171. C₄×C₆₇	C₂×H(5,𝔽₂)	semidirect
172. C₂×#3498	C₆₇	direct (N × H)
175. C₂×C₆₇	#3498	direct (N × H)
178. C₂²×H(5,𝔽₂)	C₂×C₆₇	semidirect
179. #3498	C₂×C₆₇	direct (N × H)
180. C₂×#550	C₂×C₆₇	semidirect
181. C₂×D₄²	C₂×C₆₇	semidirect
182. C₂×#535	C₂×C₆₇	semidirect
183. C₂×#532	C₂×C₆₇	semidirect
184. C₂²×#127	C₂×C₆₇	semidirect
185. C₂×C₄×#41	C₂×C₆₇	semidirect
186. C₆₇	C₂×#3498	direct (N × H)
188. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
190. C₂²×#41	C₂²×C₆₇	semidirect
191. C₂²×#41	C₂²×C₆₇	semidirect
192. C₂³×D₄	C₂²×C₆₇	semidirect
193. C₂³×D₄	C₂²×C₆₇	semidirect
194. C₂³×D₄	C₂²×C₆₇	semidirect
195. #550	C₂²×C₆₇	semidirect
196. D₄²	C₂²×C₆₇	semidirect
197. #535	C₂²×C₆₇	semidirect
198. #532	C₂²×C₆₇	semidirect
199. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
200. C₂×#127	C₂²×C₆₇	semidirect
201. C₂×#127	C₂²×C₆₇	semidirect
202. C₂×#127	C₂²×C₆₇	semidirect
203. C₂×#127	C₂²×C₆₇	semidirect
204. C₂×#124	C₂²×C₆₇	semidirect
205. C₂×#121	C₂²×C₆₇	semidirect
206. C₂×#121	C₂²×C₆₇	semidirect
207. C₂×#120	C₂²×C₆₇	semidirect
208. C₄×#41	C₂²×C₆₇	semidirect
209. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
210. C₂×C₄×D₄	C₂²×C₆₇	semidirect
211. C₂×C₄×D₄	C₂²×C₆₇	semidirect
212. C₂×#117	C₂²×C₆₇	semidirect
213. C₂²×C₄²	C₂²×C₆₇	semidirect
218. C₂×#41	D₄×C₆₇	semidirect
220. C₂×#41	C₂³×C₆₇	semidirect
221. C₂×#41	C₂³×C₆₇	semidirect
233. C₂²×D₄	C₂³×C₆₇	semidirect
234. C₂²×D₄	C₂³×C₆₇	semidirect
235. C₂²×D₄	C₂³×C₆₇	semidirect
239. C₄⋊Q₈	C₂³×C₆₇	semidirect
240. #127	C₂³×C₆₇	semidirect
241. #127	C₂³×C₆₇	semidirect
242. #127	C₂³×C₆₇	semidirect
243. #124	C₂³×C₆₇	semidirect
244. #121	C₂³×C₆₇	semidirect
245. #121	C₂³×C₆₇	semidirect
246. #120	C₂³×C₆₇	semidirect
247. C₄×Q₈	C₂³×C₆₇	semidirect
248. C₄×D₄	C₂³×C₆₇	semidirect
249. C₄×D₄	C₂³×C₆₇	semidirect
250. #117	C₂³×C₆₇	semidirect
258. C₂×C₄²	C₂³×C₆₇	semidirect
266. #41	C₂×D₄×C₆₇	semidirect
268. C₂×Q₈	C₂×D₄×C₆₇	semidirect
279. C₂×D₄	C₂×D₄×C₆₇	semidirect
288. C₂²×C₄	C₂×D₄×C₆₇	semidirect
315. Q₈	C₂²×D₄×C₆₇	semidirect
321. D₄	C₂²×D₄×C₆₇	semidirect
328. C₂×C₄	C₂²×D₄×C₆₇	semidirect
329. C₂×C₄	H(5,𝔽₂)×C₆₇	semidirect
349. C₄	C₂×H(5,𝔽₂)×C₆₇	semidirect
352. C₂	C₆₇×#3498	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×H(5,𝔽₂)×C₆₇	2
C₆₇×#3498	2
C₂×#550×C₆₇	2
C₂×D₄²×C₆₇	2
C₂×#535×C₆₇	2
C₂×#532×C₆₇	2
C₂²×#127×C₆₇	2
C₂×C₄×#41×C₆₇	2
C₂×#3498	67

Subgroups — C2×C67×#3498 #30742046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3498 #30742046